Solve the System of Equations y=3x+5 y=4x^2+x

$y = 3 x + 5$ $y = 4 x^{2} + x$

Step 1

Eliminate the equal sides of each equation and combine.

$3 x + 5 = 4 x^{2} + x$

Step 2

Solve $3 x + 5 = 4 x^{2} + x$ for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 x^{2} + x = 3 x + 5$

Move all terms containing $x$ to the left side of the equation.

Subtract $3 x$ from both sides of the equation.

$4 x^{2} + x - 3 x = 5$

Subtract $3 x$ from $x$ .

$4 x^{2} - 2 x = 5$

Subtract $5$ from both sides of the equation.

$4 x^{2} - 2 x - 5 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = 4 x^{2} + x$

Substitute the values $a = 4$ , $b = - 2$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (4 \cdot - 5)}}{2 \cdot 4} + y = 4 x^{2} + x$

Simplify.

Simplify the numerator.

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 4 \cdot - 5}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot - 5$ .

Multiply $- 4$ by $4$ .

$x = \frac{2 \pm \sqrt{4 - 16 \cdot - 5}}{2 \cdot 4}$

Multiply $- 16$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 80}}{2 \cdot 4}$

Add $4$ and $80$ .

$x = \frac{2 \pm \sqrt{84}}{2 \cdot 4}$

Rewrite $84$ as $2^{2} \cdot 21$ .

Factor $4$ out of $84$ .

$x = \frac{2 \pm \sqrt{4 (21)}}{2 \cdot 4}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{21}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{2 \pm 2 \sqrt{21}}{8}$

Simplify $\frac{2 \pm 2 \sqrt{21}}{8}$ .

$x = \frac{1 \pm \sqrt{21}}{4}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 4 \cdot - 5}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot - 5$ .

Multiply $- 4$ by $4$ .

$x = \frac{2 \pm \sqrt{4 - 16 \cdot - 5}}{2 \cdot 4}$

Multiply $- 16$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 80}}{2 \cdot 4}$

Add $4$ and $80$ .

$x = \frac{2 \pm \sqrt{84}}{2 \cdot 4}$

Rewrite $84$ as $2^{2} \cdot 21$ .

Factor $4$ out of $84$ .

$x = \frac{2 \pm \sqrt{4 (21)}}{2 \cdot 4}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{21}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{2 \pm 2 \sqrt{21}}{8}$

Simplify $\frac{2 \pm 2 \sqrt{21}}{8}$ .

$x = \frac{1 \pm \sqrt{21}}{4}$

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{21}}{4}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 4 \cdot - 5}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot - 5$ .

Multiply $- 4$ by $4$ .

$x = \frac{2 \pm \sqrt{4 - 16 \cdot - 5}}{2 \cdot 4}$

Multiply $- 16$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 80}}{2 \cdot 4}$

Add $4$ and $80$ .

$x = \frac{2 \pm \sqrt{84}}{2 \cdot 4}$

Rewrite $84$ as $2^{2} \cdot 21$ .

Factor $4$ out of $84$ .

$x = \frac{2 \pm \sqrt{4 (21)}}{2 \cdot 4}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{21}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{2 \pm 2 \sqrt{21}}{8}$

Simplify $\frac{2 \pm 2 \sqrt{21}}{8}$ .

$x = \frac{1 \pm \sqrt{21}}{4}$

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{21}}{4}$

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{21}}{4}, \frac{1 - \sqrt{21}}{4}$

Step 3

Evaluate $y$ when $x = \frac{1 + \sqrt{21}}{4}$ .

Substitute $\frac{1 + \sqrt{21}}{4}$ for $x$ .

$y = 4 {(\frac{1 + \sqrt{21}}{4})}^{2} + \frac{1 + \sqrt{21}}{4}$

Substitute $\frac{1 + \sqrt{21}}{4}$ for $x$ in $y = 4 {(\frac{1 + \sqrt{21}}{4})}^{2} + \frac{1 + \sqrt{21}}{4}$ and solve for $y$ .

Remove parentheses.

$y = 4 {(\frac{1 + \sqrt{21}}{4})}^{2} + \frac{1 + \sqrt{21}}{4}$

Simplify $4 {(\frac{1 + \sqrt{21}}{4})}^{2} + \frac{1 + \sqrt{21}}{4}$ .

Simplify each term.

Apply the product rule to $\frac{1 + \sqrt{21}}{4}$ .

$y = 4 \frac{{(1 + \sqrt{21})}^{2}}{4^{2}} + \frac{1 + \sqrt{21}}{4}$

Raise $4$ to the power of $2$ .

$y = 4 \frac{{(1 + \sqrt{21})}^{2}}{16} + \frac{1 + \sqrt{21}}{4}$

Cancel the common factor of $4$ .

Factor $4$ out of $16$ .

$y = 4 \frac{{(1 + \sqrt{21})}^{2}}{4 (4)} + \frac{1 + \sqrt{21}}{4}$

Cancel the common factor.

$y = 4 \frac{{(1 + \sqrt{21})}^{2}}{4 \cdot 4} + \frac{1 + \sqrt{21}}{4}$

Rewrite the expression.

$y = \frac{{(1 + \sqrt{21})}^{2}}{4} + \frac{1 + \sqrt{21}}{4}$

Rewrite ${(1 + \sqrt{21})}^{2}$ as $(1 + \sqrt{21}) (1 + \sqrt{21})$ .

$y = \frac{(1 + \sqrt{21}) (1 + \sqrt{21})}{4} + \frac{1 + \sqrt{21}}{4}$

Expand $(1 + \sqrt{21}) (1 + \sqrt{21})$ using the FOIL Method.

Apply the distributive property.

$y = \frac{1 (1 + \sqrt{21}) + \sqrt{21} (1 + \sqrt{21})}{4} + \frac{1 + \sqrt{21}}{4}$

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 \sqrt{21} + \sqrt{21} (1 + \sqrt{21})}{4} + \frac{1 + \sqrt{21}}{4}$

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 \sqrt{21} + \sqrt{21} \cdot 1 + \sqrt{21} \sqrt{21}}{4} + \frac{1 + \sqrt{21}}{4}$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$y = \frac{1 + 1 \sqrt{21} + \sqrt{21} \cdot 1 + \sqrt{21} \sqrt{21}}{4} + \frac{1 + \sqrt{21}}{4}$

Multiply $\sqrt{21}$ by $1$ .

$y = \frac{1 + \sqrt{21} + \sqrt{21} \cdot 1 + \sqrt{21} \sqrt{21}}{4} + \frac{1 + \sqrt{21}}{4}$

Multiply $\sqrt{21}$ by $1$ .

$y = \frac{1 + \sqrt{21} + \sqrt{21} + \sqrt{21} \sqrt{21}}{4} + \frac{1 + \sqrt{21}}{4}$

Combine using the product rule for radicals.

$y = \frac{1 + \sqrt{21} + \sqrt{21} + \sqrt{21 \cdot 21}}{4} + \frac{1 + \sqrt{21}}{4}$

Multiply $21$ by $21$ .

$y = \frac{1 + \sqrt{21} + \sqrt{21} + \sqrt{441}}{4} + \frac{1 + \sqrt{21}}{4}$

Rewrite $441$ as $21^{2}$ .

$y = \frac{1 + \sqrt{21} + \sqrt{21} + \sqrt{21^{2}}}{4} + \frac{1 + \sqrt{21}}{4}$

Pull terms out from under the radical, assuming positive real numbers.

$y = \frac{1 + \sqrt{21} + \sqrt{21} + 21}{4} + \frac{1 + \sqrt{21}}{4}$

Add $1$ and $21$ .

$y = \frac{22 + \sqrt{21} + \sqrt{21}}{4} + \frac{1 + \sqrt{21}}{4}$

Add $\sqrt{21}$ and $\sqrt{21}$ .

$y = \frac{22 + 2 \sqrt{21}}{4} + \frac{1 + \sqrt{21}}{4}$

Cancel the common factor of $22 + 2 \sqrt{21}$ and $4$ .

Factor $2$ out of $22$ .

$y = \frac{2 \cdot 11 + 2 \sqrt{21}}{4} + \frac{1 + \sqrt{21}}{4}$

Factor $2$ out of $2 \sqrt{21}$ .

$y = \frac{2 \cdot 11 + 2 (\sqrt{21})}{4} + \frac{1 + \sqrt{21}}{4}$

Factor $2$ out of $2 \cdot 11 + 2 (\sqrt{21})$ .

$y = \frac{2 \cdot (11 + \sqrt{21})}{4} + \frac{1 + \sqrt{21}}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{2 \cdot (11 + \sqrt{21})}{2 (2)} + \frac{1 + \sqrt{21}}{4}$

Cancel the common factor.

$y = \frac{2 \cdot (11 + \sqrt{21})}{2 \cdot 2} + \frac{1 + \sqrt{21}}{4}$

Rewrite the expression.

$y = \frac{11 + \sqrt{21}}{2} + \frac{1 + \sqrt{21}}{4}$

To write $\frac{11 + \sqrt{21}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{11 + \sqrt{21}}{2} \cdot \frac{2}{2} + \frac{1 + \sqrt{21}}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{11 + \sqrt{21}}{2}$ by $\frac{2}{2}$ .

$y = \frac{(11 + \sqrt{21}) \cdot 2}{2 \cdot 2} + \frac{1 + \sqrt{21}}{4}$

Multiply $2$ by $2$ .

$y = \frac{(11 + \sqrt{21}) \cdot 2}{4} + \frac{1 + \sqrt{21}}{4}$

Combine the numerators over the common denominator.

$y = \frac{(11 + \sqrt{21}) \cdot 2 + 1 + \sqrt{21}}{4}$

Simplify the numerator.

Apply the distributive property.

$y = \frac{11 \cdot 2 + \sqrt{21} \cdot 2 + 1 + \sqrt{21}}{4}$

Multiply $11$ by $2$ .

$y = \frac{22 + \sqrt{21} \cdot 2 + 1 + \sqrt{21}}{4}$

Move $2$ to the left of $\sqrt{21}$ .

$y = \frac{22 + 2 \cdot \sqrt{21} + 1 + \sqrt{21}}{4}$

Add $22$ and $1$ .

$y = \frac{23 + 2 \sqrt{21} + \sqrt{21}}{4}$

Add $2 \sqrt{21}$ and $\sqrt{21}$ .

$y = \frac{23 + 3 \sqrt{21}}{4}$

Step 4

Evaluate $y$ when $x = \frac{1 - \sqrt{21}}{4}$ .

Substitute $\frac{1 - \sqrt{21}}{4}$ for $x$ .

$y = 4 {(\frac{1 - \sqrt{21}}{4})}^{2} + \frac{1 - \sqrt{21}}{4}$

Substitute $\frac{1 - \sqrt{21}}{4}$ for $x$ in $y = 4 {(\frac{1 - \sqrt{21}}{4})}^{2} + \frac{1 - \sqrt{21}}{4}$ and solve for $y$ .

Remove parentheses.

$y = 4 {(\frac{1 - \sqrt{21}}{4})}^{2} + \frac{1 - \sqrt{21}}{4}$

Simplify $4 {(\frac{1 - \sqrt{21}}{4})}^{2} + \frac{1 - \sqrt{21}}{4}$ .

Simplify each term.

Apply the product rule to $\frac{1 - \sqrt{21}}{4}$ .

$y = 4 \frac{{(1 - \sqrt{21})}^{2}}{4^{2}} + \frac{1 - \sqrt{21}}{4}$

Raise $4$ to the power of $2$ .

$y = 4 \frac{{(1 - \sqrt{21})}^{2}}{16} + \frac{1 - \sqrt{21}}{4}$

Cancel the common factor of $4$ .

Factor $4$ out of $16$ .

$y = 4 \frac{{(1 - \sqrt{21})}^{2}}{4 (4)} + \frac{1 - \sqrt{21}}{4}$

Cancel the common factor.

$y = 4 \frac{{(1 - \sqrt{21})}^{2}}{4 \cdot 4} + \frac{1 - \sqrt{21}}{4}$

Rewrite the expression.

$y = \frac{{(1 - \sqrt{21})}^{2}}{4} + \frac{1 - \sqrt{21}}{4}$

Rewrite ${(1 - \sqrt{21})}^{2}$ as $(1 - \sqrt{21}) (1 - \sqrt{21})$ .

$y = \frac{(1 - \sqrt{21}) (1 - \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Expand $(1 - \sqrt{21}) (1 - \sqrt{21})$ using the FOIL Method.

Apply the distributive property.

$y = \frac{1 (1 - \sqrt{21}) - \sqrt{21} (1 - \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 (- \sqrt{21}) - \sqrt{21} (1 - \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 (- \sqrt{21}) - \sqrt{21} \cdot 1 - \sqrt{21} (- \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$y = \frac{1 + 1 (- \sqrt{21}) - \sqrt{21} \cdot 1 - \sqrt{21} (- \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Multiply $- \sqrt{21}$ by $1$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} \cdot 1 - \sqrt{21} (- \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Multiply $- 1$ by $1$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} - \sqrt{21} (- \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Multiply $- \sqrt{21} (- \sqrt{21})$ .

Multiply $- 1$ by $- 1$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} + 1 \sqrt{21} \sqrt{21}}{4} + \frac{1 - \sqrt{21}}{4}$

Multiply $\sqrt{21}$ by $1$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} + \sqrt{21} \sqrt{21}}{4} + \frac{1 - \sqrt{21}}{4}$

Raise $\sqrt{21}$ to the power of $1$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} + {\sqrt{21}}^{1} \sqrt{21}}{4} + \frac{1 - \sqrt{21}}{4}$

Raise $\sqrt{21}$ to the power of $1$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} + {\sqrt{21}}^{1} {\sqrt{21}}^{1}}{4} + \frac{1 - \sqrt{21}}{4}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 - \sqrt{21} - \sqrt{21} + {\sqrt{21}}^{1 + 1}}{4} + \frac{1 - \sqrt{21}}{4}$

Add $1$ and $1$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} + {\sqrt{21}}^{2}}{4} + \frac{1 - \sqrt{21}}{4}$

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} + {(21^{\frac{1}{2}})}^{2}}{4} + \frac{1 - \sqrt{21}}{4}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} + 21^{\frac{1}{2} \cdot 2}}{4} + \frac{1 - \sqrt{21}}{4}$

Combine $\frac{1}{2}$ and $2$ .

$y = \frac{1 - \sqrt{21} - \sqrt{21} + 21^{\frac{2}{2}}}{4} + \frac{1 - \sqrt{21}}{4}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{1 - \sqrt{21} - \sqrt{21} + 21^{\frac{2}{2}}}{4} + \frac{1 - \sqrt{21}}{4}$

Rewrite the expression.

$y = \frac{1 - \sqrt{21} - \sqrt{21} + 21^{1}}{4} + \frac{1 - \sqrt{21}}{4}$

Evaluate the exponent.

$y = \frac{1 - \sqrt{21} - \sqrt{21} + 21}{4} + \frac{1 - \sqrt{21}}{4}$

Add $1$ and $21$ .

$y = \frac{22 - \sqrt{21} - \sqrt{21}}{4} + \frac{1 - \sqrt{21}}{4}$

Subtract $\sqrt{21}$ from $- \sqrt{21}$ .

$y = \frac{22 - 2 \sqrt{21}}{4} + \frac{1 - \sqrt{21}}{4}$

Cancel the common factor of $22 - 2 \sqrt{21}$ and $4$ .

Factor $2$ out of $22$ .

$y = \frac{2 (11) - 2 \sqrt{21}}{4} + \frac{1 - \sqrt{21}}{4}$

Factor $2$ out of $- 2 \sqrt{21}$ .

$y = \frac{2 (11) + 2 (- \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Factor $2$ out of $2 (11) + 2 (- \sqrt{21})$ .

$y = \frac{2 (11 - \sqrt{21})}{4} + \frac{1 - \sqrt{21}}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{2 (11 - \sqrt{21})}{2 \cdot 2} + \frac{1 - \sqrt{21}}{4}$

Cancel the common factor.

$y = \frac{2 (11 - \sqrt{21})}{2 \cdot 2} + \frac{1 - \sqrt{21}}{4}$

Rewrite the expression.

$y = \frac{11 - \sqrt{21}}{2} + \frac{1 - \sqrt{21}}{4}$

To write $\frac{11 - \sqrt{21}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{11 - \sqrt{21}}{2} \cdot \frac{2}{2} + \frac{1 - \sqrt{21}}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{11 - \sqrt{21}}{2}$ by $\frac{2}{2}$ .

$y = \frac{(11 - \sqrt{21}) \cdot 2}{2 \cdot 2} + \frac{1 - \sqrt{21}}{4}$

Multiply $2$ by $2$ .

$y = \frac{(11 - \sqrt{21}) \cdot 2}{4} + \frac{1 - \sqrt{21}}{4}$

Combine the numerators over the common denominator.

$y = \frac{(11 - \sqrt{21}) \cdot 2 + 1 - \sqrt{21}}{4}$

Simplify the numerator.

Apply the distributive property.

$y = \frac{11 \cdot 2 - \sqrt{21} \cdot 2 + 1 - \sqrt{21}}{4}$

Multiply $11$ by $2$ .

$y = \frac{22 - \sqrt{21} \cdot 2 + 1 - \sqrt{21}}{4}$

Multiply $2$ by $- 1$ .

$y = \frac{22 - 2 \sqrt{21} + 1 - \sqrt{21}}{4}$

Add $22$ and $1$ .

$y = \frac{23 - 2 \sqrt{21} - \sqrt{21}}{4}$

Subtract $\sqrt{21}$ from $- 2 \sqrt{21}$ .

$y = \frac{23 - 3 \sqrt{21}}{4}$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(\frac{1 + \sqrt{21}}{4}, \frac{23 + 3 \sqrt{21}}{4})$

$(\frac{1 - \sqrt{21}}{4}, \frac{23 - 3 \sqrt{21}}{4})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{1 + \sqrt{21}}{4}, \frac{23 + 3 \sqrt{21}}{4}), (\frac{1 - \sqrt{21}}{4}, \frac{23 - 3 \sqrt{21}}{4})$

Equation Form:

$x = \frac{1 + \sqrt{21}}{4}, y = \frac{23 + 3 \sqrt{21}}{4}$

$x = \frac{1 - \sqrt{21}}{4}, y = \frac{23 - 3 \sqrt{21}}{4}$

Step 7

Solve the System of Equations y=3x+5 y=4x^2+x

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