Solve the System of Equations f(x)=x^2-2x+3 , f(x)=-2x+7

$f (x) = x^{2} - 2 x + 3$ , $f (x) = - 2 x + 7$

Step 1

Eliminate the equal sides of each equation and combine.

$x^{2} - 2 x + 3 = - 2 x + 7$

Step 2

Solve $x^{2} - 2 x + 3 = - 2 x + 7$ for $x$ .

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

Add $2 x$ to both sides of the equation.

$x^{2} - 2 x + 3 + 2 x = 7$

Combine the opposite terms in $x^{2} - 2 x + 3 + 2 x$ .

Add $- 2 x$ and $2 x$ .

$x^{2} + 0 + 3 = 7$

Add $x^{2}$ and $0$ .

$x^{2} + 3 = 7$

Move all terms not containing $x$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$x^{2} = 7 - 3$

Subtract $3$ from $7$ .

$x^{2} = 4$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Simplify $\pm \sqrt{4}$ .

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

$x = \pm \sqrt{2^{2}}$

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

$x = \pm 2$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Step 3

Evaluate $f$ when $x = 2$ .

Substitute $2$ for $x$ .

$f (2) = - 2 \cdot 2 + 7$

Substitute $2$ for $x$ in $f (2) = - 2 \cdot 2 + 7$ and solve for $f$ .

Simplify.

Multiply $- 2$ by $2$ .

$f (2) = - 4 + 7$

Add $- 4$ and $7$ .

$f (2) = 3$

Divide each term in $f (2) = 3$ by $2$ and simplify.

Divide each term in $f (2) = 3$ by $2$ .

$\frac{f (2)}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{f \cdot 2}{2} = \frac{3}{2}$

Divide $f$ by $1$ .

$f = \frac{3}{2}$

Step 4

Evaluate $f$ when $x = - 2$ .

Substitute $- 2$ for $x$ .

$f (- 2) = - 2 \cdot - 2 + 7$

Substitute $- 2$ for $x$ in $f (- 2) = - 2 \cdot - 2 + 7$ and solve for $f$ .

Simplify.

Multiply $- 2$ by $- 2$ .

$f (- 2) = 4 + 7$

Add $4$ and $7$ .

$f (- 2) = 11$

Divide each term in $f (- 2) = 11$ by $- 2$ and simplify.

Divide each term in $f (- 2) = 11$ by $- 2$ .

$\frac{f (- 2)}{- 2} = \frac{11}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{f \cdot - 2}{- 2} = \frac{11}{- 2}$

Divide $f$ by $1$ .

$f = \frac{11}{- 2}$

Simplify the right side.

Move the negative in front of the fraction.

$f = - \frac{11}{2}$

Step 5

The solution of the system of equations is all the values that make the system true.

$f = \frac{3}{2}, x = 2 f = - \frac{11}{2}, x = - 2$

Step 6

List all of the solutions.

$f = \frac{3}{2}, x = 2$

$f = - \frac{11}{2}, x = - 2$

Solve the System of Equations f(x)=x^2-2x+3 , f(x)=-2x+7

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