Solve x^2+7x-2^2=0

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

Solve for x

$$x=\frac{\sqrt{65}-7}{2}\approx 0.531128874$$

$$x=\frac{-\sqrt{65}-7}{2}\approx -7.531128874$$

Steps Using the Quadratic Formula

Calculate 2 to the power of 2 and get 4.

$$x^{2}+7x-4=0$$

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

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

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-4\right)}}{2}$$

Multiply -4 times -4.

$$x=\frac{-7±\sqrt{49+16}}{2}$$

Add 49 to 16.

$$x=\frac{-7±\sqrt{65}}{2}$$

Now solve the equation x=\frac{-7±\sqrt{65}}{2} when ± is plus. Add -7 to \sqrt{65}.

$$x=\frac{\sqrt{65}-7}{2}$$

Now solve the equation x=\frac{-7±\sqrt{65}}{2} when ± is minus. Subtract \sqrt{65} from -7.

$$x=\frac{-\sqrt{65}-7}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{65}-7}{2} x=\frac{-\sqrt{65}-7}{2}$$

Steps for Completing the Square

Calculate 2 to the power of 2 and get 4.

$$x^{2}+7x-4=0$$

Add 4 to both sides. Anything plus zero gives itself.

$$x^{2}+7x=4$$

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=4+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+7x+\frac{49}{4}=4+\frac{49}{4}$$

Add 4 to \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{65}{4}$$

Factor x^{2}+7x+\frac{49}{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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{65}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{65}{4}}$$

Simplify.

$$x+\frac{7}{2}=\frac{\sqrt{65}}{2} x+\frac{7}{2}=-\frac{\sqrt{65}}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$x=\frac{\sqrt{65}-7}{2} x=\frac{-\sqrt{65}-7}{2}$$