Solve x^2-4x+4=5x-3

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

Solve for x

$$x = \frac{\sqrt{53} + 9}{2} \approx 8.140054945$$

$$x=\frac{9-\sqrt{53}}{2}\approx 0.859945055$$

Steps Using the Quadratic Formula

Subtract 5x from both sides.

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

Combine -4x and -5x to get -9x.

$$x^{2}-9x+4=-3$$

Add 3 to both sides.

$$x^{2}-9x+4+3=0$$

Add 4 and 3 to get 7.

$$x^{2}-9x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -9 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 7}}{2}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 7}}{2}$$

Multiply -4 times 7.

$$x=\frac{-\left(-9\right)±\sqrt{81-28}}{2}$$

Add 81 to -28.

$$x=\frac{-\left(-9\right)±\sqrt{53}}{2}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{53}}{2}$$

Now solve the equation x=\frac{9±\sqrt{53}}{2} when ± is plus. Add 9 to \sqrt{53}.

$$x=\frac{\sqrt{53}+9}{2}$$

Now solve the equation x=\frac{9±\sqrt{53}}{2} when ± is minus. Subtract \sqrt{53} from 9.

$$x=\frac{9-\sqrt{53}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{53}+9}{2} x=\frac{9-\sqrt{53}}{2}$$

Steps for Completing the Square

Subtract 5x from both sides.

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

Combine -4x and -5x to get -9x.

$$x^{2}-9x+4=-3$$

Subtract 4 from both sides.

$$x^{2}-9x=-3-4$$

Subtract 4 from -3 to get -7.

$$x^{2}-9x=-7$$

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.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-7+\left(-\frac{9}{2}\right)^{2}$$

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

$$x^{2}-9x+\frac{81}{4}=-7+\frac{81}{4}$$

Add -7 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{53}{4}$$

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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{53}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{53}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{\sqrt{53}}{2} x-\frac{9}{2}=-\frac{\sqrt{53}}{2}$$

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

$$x=\frac{\sqrt{53}+9}{2} x=\frac{9-\sqrt{53}}{2}$$