Solve 7x=25-x^2

$$7 x = 25 - x ^ { 2 }$$

Solve for x

$$x = \frac{\sqrt{149} - 7}{2} \approx 2.603277808$$

$$x=\frac{-\sqrt{149}-7}{2}\approx -9.603277808$$

Steps Using the Quadratic Formula

Subtract 25 from both sides.

$$7x-25=-x^{2}$$

Add x^{2} to both sides.

$$7x-25+x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+7x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 7 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-25\right)}}{2}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-25\right)}}{2}$$

Multiply -4 times -25.

$$x=\frac{-7±\sqrt{49+100}}{2}$$

Add 49 to 100.

$$x=\frac{-7±\sqrt{149}}{2}$$

Now solve the equation x=\frac{-7±\sqrt{149}}{2} when ± is plus. Add -7 to \sqrt{149}.

$$x=\frac{\sqrt{149}-7}{2}$$

Now solve the equation x=\frac{-7±\sqrt{149}}{2} when ± is minus. Subtract \sqrt{149} from -7.

$$x=\frac{-\sqrt{149}-7}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{149}-7}{2} x=\frac{-\sqrt{149}-7}{2}$$

Steps for Completing the Square

Add x^{2} to both sides.

$$7x+x^{2}=25$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+7x=25$$

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.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=25+\left(\frac{7}{2}\right)^{2}$$

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

$$x^{2}+7x+\frac{49}{4}=25+\frac{49}{4}$$

Add 25 to \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{149}{4}$$

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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{149}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{149}{4}}$$

Simplify.

$$x+\frac{7}{2}=\frac{\sqrt{149}}{2} x+\frac{7}{2}=-\frac{\sqrt{149}}{2}$$

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

$$x=\frac{\sqrt{149}-7}{2} x=\frac{-\sqrt{149}-7}{2}$$