Find the Points of Intersection y=-x^2+6x 4y=21-x

$y = - x^{2} + 6 x$ $4 y = 21 - x$

Step 1

Replace all occurrences of $y$ with $- x^{2} + 6 x$ in each equation.

Replace all occurrences of $y$ in $4 y = 21 - x$ with $- x^{2} + 6 x$ .

$4 (- x^{2} + 6 x) = 21 - x$

$y = - x^{2} + 6 x$

Simplify the left side.

Simplify $4 (- x^{2} + 6 x)$ .

Apply the distributive property.

$4 (- x^{2}) + 4 (6 x) = 21 - x$

$y = - x^{2} + 6 x$

Multiply.

Multiply $- 1$ by $4$ .

$- 4 x^{2} + 4 (6 x) = 21 - x$

$y = - x^{2} + 6 x$

Multiply $6$ by $4$ .

$- 4 x^{2} + 24 x = 21 - x$

$y = - x^{2} + 6 x$

$- 4 x^{2} + 24 x = 21 - x$

$y = - x^{2} + 6 x$

$- 4 x^{2} + 24 x = 21 - x$

$y = - x^{2} + 6 x$

$- 4 x^{2} + 24 x = 21 - x$

$y = - x^{2} + 6 x$

$- 4 x^{2} + 24 x = 21 - x$

$y = - x^{2} + 6 x$

Step 2

Solve for $x$ in $- 4 x^{2} + 24 x = 21 - x$ .

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

Add $x$ to both sides of the equation.

$- 4 x^{2} + 24 x + x = 21$

$y = - x^{2} + 6 x$

Add $24 x$ and $x$ .

$- 4 x^{2} + 25 x = 21$

$y = - x^{2} + 6 x$

$- 4 x^{2} + 25 x = 21$

$y = - x^{2} + 6 x$

Subtract $21$ from both sides of the equation.

$- 4 x^{2} + 25 x - 21 = 0$

$y = - x^{2} + 6 x$

Factor the left side of the equation.

Factor $- 1$ out of $- 4 x^{2} + 25 x - 21$ .

Factor $- 1$ out of $- 4 x^{2}$ .

$- (4 x^{2}) + 25 x - 21 = 0$

$y = - x^{2} + 6 x$

Factor $- 1$ out of $25 x$ .

$- (4 x^{2}) - (- 25 x) - 21 = 0$

$y = - x^{2} + 6 x$

Rewrite $- 21$ as $- 1 (21)$ .

$- (4 x^{2}) - (- 25 x) - 1 \cdot 21 = 0$

$y = - x^{2} + 6 x$

Factor $- 1$ out of $- (4 x^{2}) - (- 25 x)$ .

$- (4 x^{2} - 25 x) - 1 \cdot 21 = 0$

$y = - x^{2} + 6 x$

Factor $- 1$ out of $- (4 x^{2} - 25 x) - 1 (21)$ .

$- (4 x^{2} - 25 x + 21) = 0$

$y = - x^{2} + 6 x$

$- (4 x^{2} - 25 x + 21) = 0$

$y = - x^{2} + 6 x$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 21 = 84$ and whose sum is $b = - 25$ .

Factor $- 25$ out of $- 25 x$ .

$- (4 x^{2} - 25 x + 21) = 0$

$y = - x^{2} + 6 x$

Rewrite $- 25$ as $- 4$ plus $- 21$

$- (4 x^{2} + (- 4 - 21) x + 21) = 0$

$y = - x^{2} + 6 x$

Apply the distributive property.

$- (4 x^{2} - 4 x - 21 x + 21) = 0$

$y = - x^{2} + 6 x$

$- (4 x^{2} - 4 x - 21 x + 21) = 0$

$y = - x^{2} + 6 x$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- ((4 x^{2} - 4 x) - 21 x + 21) = 0$

$y = - x^{2} + 6 x$

Factor out the greatest common factor (GCF) from each group.

$- (4 x (x - 1) - 21 (x - 1)) = 0$

$y = - x^{2} + 6 x$

$- (4 x (x - 1) - 21 (x - 1)) = 0$

$y = - x^{2} + 6 x$

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$- ((x - 1) (4 x - 21)) = 0$

$y = - x^{2} + 6 x$

$- ((x - 1) (4 x - 21)) = 0$

$y = - x^{2} + 6 x$

Remove unnecessary parentheses.

$- (x - 1) (4 x - 21) = 0$

$y = - x^{2} + 6 x$

$- (x - 1) (4 x - 21) = 0$

$y = - x^{2} + 6 x$

$- (x - 1) (4 x - 21) = 0$

$y = - x^{2} + 6 x$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$4 x - 21 = 0$

$y = - x^{2} + 6 x$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

$y = - x^{2} + 6 x$

Add $1$ to both sides of the equation.

$x = 1$

$y = - x^{2} + 6 x$

$x = 1$

$y = - x^{2} + 6 x$

Set $4 x - 21$ equal to $0$ and solve for $x$ .

Set $4 x - 21$ equal to $0$ .

$4 x - 21 = 0$

$y = - x^{2} + 6 x$

Solve $4 x - 21 = 0$ for $x$ .

Add $21$ to both sides of the equation.

$4 x = 21$

$y = - x^{2} + 6 x$

Divide each term in $4 x = 21$ by $4$ and simplify.

Divide each term in $4 x = 21$ by $4$ .

$\frac{4 x}{4} = \frac{21}{4}$

$y = - x^{2} + 6 x$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{21}{4}$

$y = - x^{2} + 6 x$

Divide $x$ by $1$ .

$x = \frac{21}{4}$

$y = - x^{2} + 6 x$

$x = \frac{21}{4}$

$y = - x^{2} + 6 x$

$x = \frac{21}{4}$

$y = - x^{2} + 6 x$

$x = \frac{21}{4}$

$y = - x^{2} + 6 x$

$x = \frac{21}{4}$

$y = - x^{2} + 6 x$

$x = \frac{21}{4}$

$y = - x^{2} + 6 x$

The final solution is all the values that make $- (x - 1) (4 x - 21) = 0$ true.

$x = 1, \frac{21}{4}$

$y = - x^{2} + 6 x$

$x = 1, \frac{21}{4}$

$y = - x^{2} + 6 x$

Step 3

Replace all occurrences of $x$ with $1$ in each equation.

Replace all occurrences of $x$ in $y = - x^{2} + 6 x$ with $1$ .

$y = - {(1)}^{2} + 6 (1)$

$x = 1$

Simplify the right side.

Simplify $- {(1)}^{2} + 6 (1)$ .

Simplify each term.

One to any power is one.

$y = - 1 \cdot 1 + 6 (1)$

$x = 1$

Multiply $- 1$ by $1$ .

$y = - 1 + 6 (1)$

$x = 1$

Multiply $6$ by $1$ .

$y = - 1 + 6$

$x = 1$

$y = - 1 + 6$

$x = 1$

Add $- 1$ and $6$ .

$y = 5$

$x = 1$

$y = 5$

$x = 1$

$y = 5$

$x = 1$

$y = 5$

$x = 1$

Step 4

Replace all occurrences of $x$ with $\frac{21}{4}$ in each equation.

Replace all occurrences of $x$ in $y = - x^{2} + 6 x$ with $\frac{21}{4}$ .

$y = - {(\frac{21}{4})}^{2} + 6 (\frac{21}{4})$

$x = \frac{21}{4}$

Simplify the right side.

Simplify $- {(\frac{21}{4})}^{2} + 6 (\frac{21}{4})$ .

Simplify each term.

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

$y = - \frac{21^{2}}{4^{2}} + 6 (\frac{21}{4})$

$x = \frac{21}{4}$

Raise $21$ to the power of $2$ .

$y = - \frac{441}{4^{2}} + 6 (\frac{21}{4})$

$x = \frac{21}{4}$

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

$y = - \frac{441}{16} + 6 (\frac{21}{4})$

$x = \frac{21}{4}$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$y = - \frac{441}{16} + 2 (3) (\frac{21}{4})$

$x = \frac{21}{4}$

Factor $2$ out of $4$ .

$y = - \frac{441}{16} + 2 \cdot (3 (\frac{21}{2 \cdot 2}))$

$x = \frac{21}{4}$

Cancel the common factor.

$y = - \frac{441}{16} + 2 \cdot (3 (\frac{21}{2 \cdot 2}))$

$x = \frac{21}{4}$

Rewrite the expression.

$y = - \frac{441}{16} + 3 (\frac{21}{2})$

$x = \frac{21}{4}$

$y = - \frac{441}{16} + 3 (\frac{21}{2})$

$x = \frac{21}{4}$

Combine $3$ and $\frac{21}{2}$ .

$y = - \frac{441}{16} + \frac{3 \cdot 21}{2}$

$x = \frac{21}{4}$

Multiply $3$ by $21$ .

$y = - \frac{441}{16} + \frac{63}{2}$

$x = \frac{21}{4}$

$y = - \frac{441}{16} + \frac{63}{2}$

$x = \frac{21}{4}$

To write $\frac{63}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = - \frac{441}{16} + \frac{63}{2} \cdot \frac{8}{8}$

$x = \frac{21}{4}$

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{63}{2}$ by $\frac{8}{8}$ .

$y = - \frac{441}{16} + \frac{63 \cdot 8}{2 \cdot 8}$

$x = \frac{21}{4}$

Multiply $2$ by $8$ .

$y = - \frac{441}{16} + \frac{63 \cdot 8}{16}$

$x = \frac{21}{4}$

$y = - \frac{441}{16} + \frac{63 \cdot 8}{16}$

$x = \frac{21}{4}$

Combine the numerators over the common denominator.

$y = \frac{- 441 + 63 \cdot 8}{16}$

$x = \frac{21}{4}$

Simplify the numerator.

Multiply $63$ by $8$ .

$y = \frac{- 441 + 504}{16}$

$x = \frac{21}{4}$

Add $- 441$ and $504$ .

$y = \frac{63}{16}$

$x = \frac{21}{4}$

$y = \frac{63}{16}$

$x = \frac{21}{4}$

$y = \frac{63}{16}$

$x = \frac{21}{4}$

$y = \frac{63}{16}$

$x = \frac{21}{4}$

$y = \frac{63}{16}$

$x = \frac{21}{4}$

Step 5

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

$(1, 5)$

$(\frac{21}{4}, \frac{63}{16})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(1, 5), (\frac{21}{4}, \frac{63}{16})$

Equation Form:

$x = 1, y = 5$

$x = \frac{21}{4}, y = \frac{63}{16}$

Step 7

Find the Points of Intersection y=-x^2+6x 4y=21-x

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