Solve x^2-2x+3=7x-8

$$x ^ { 2 } - 2 x + 3 = 7 x - 8$$

Solve for x

$$x = \frac{\sqrt{37} + 9}{2} \approx 7.541381265$$

$$x = \frac{9 - \sqrt{37}}{2} \approx 1.458618735$$

Steps Using the Quadratic Formula

Subtract 7x from both sides.

$$x^{2}-2x+3-7x=-8$$

Combine -2x and -7x to get -9x.

$$x^{2}-9x+3=-8$$

Add 8 to both sides.

$$x^{2}-9x+3+8=0$$

Add 3 and 8 to get 11.

$$x^{2}-9x+11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -9 for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 11}}{2}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 11}}{2}$$

Multiply -4 times 11.

$$x=\frac{-\left(-9\right)±\sqrt{81-44}}{2}$$

Add 81 to -44.

$$x=\frac{-\left(-9\right)±\sqrt{37}}{2}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{37}}{2}$$

Now solve the equation x=\frac{9±\sqrt{37}}{2} when ± is plus. Add 9 to \sqrt{37}.

$$x=\frac{\sqrt{37}+9}{2}$$

Now solve the equation x=\frac{9±\sqrt{37}}{2} when ± is minus. Subtract \sqrt{37} from 9.

$$x=\frac{9-\sqrt{37}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{37}+9}{2} x=\frac{9-\sqrt{37}}{2}$$

Steps for Completing the Square

Subtract 7x from both sides.

$$x^{2}-2x+3-7x=-8$$

Combine -2x and -7x to get -9x.

$$x^{2}-9x+3=-8$$

Subtract 3 from both sides.

$$x^{2}-9x=-8-3$$

Subtract 3 from -8 to get -11.

$$x^{2}-9x=-11$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-11+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=-11+\frac{81}{4}$$

Add -11 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{37}{4}$$

Factor x^{2}-9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{37}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{37}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{\sqrt{37}}{2} x-\frac{9}{2}=-\frac{\sqrt{37}}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{\sqrt{37}+9}{2} x=\frac{9-\sqrt{37}}{2}$$