Solve 7x=4x(x+30)+30

$$7 x = 4 x ( x + 30 ) + 30$$

Solve for x

$$x=\frac{\sqrt{12289}-113}{8}\approx -0.268029732$$

$$x=\frac{-\sqrt{12289}-113}{8}\approx -27.981970268$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x by x+30.

$$7x=4x^{2}+120x+30$$

Subtract 4x^{2} from both sides.

$$7x-4x^{2}=120x+30$$

Subtract 120x from both sides.

$$7x-4x^{2}-120x=30$$

Combine 7x and -120x to get -113x.

$$-113x-4x^{2}=30$$

Subtract 30 from both sides.

$$-113x-4x^{2}-30=0$$

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.

$$-4x^{2}-113x-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -113 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-113\right)±\sqrt{\left(-113\right)^{2}-4\left(-4\right)\left(-30\right)}}{2\left(-4\right)}$$

Square -113.

$$x=\frac{-\left(-113\right)±\sqrt{12769-4\left(-4\right)\left(-30\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-113\right)±\sqrt{12769+16\left(-30\right)}}{2\left(-4\right)}$$

Multiply 16 times -30.

$$x=\frac{-\left(-113\right)±\sqrt{12769-480}}{2\left(-4\right)}$$

Add 12769 to -480.

$$x=\frac{-\left(-113\right)±\sqrt{12289}}{2\left(-4\right)}$$

The opposite of -113 is 113.

$$x=\frac{113±\sqrt{12289}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{113±\sqrt{12289}}{-8}$$

Now solve the equation x=\frac{113±\sqrt{12289}}{-8} when ± is plus. Add 113 to \sqrt{12289}.

$$x=\frac{\sqrt{12289}+113}{-8}$$

Divide 113+\sqrt{12289} by -8.

$$x=\frac{-\sqrt{12289}-113}{8}$$

Now solve the equation x=\frac{113±\sqrt{12289}}{-8} when ± is minus. Subtract \sqrt{12289} from 113.

$$x=\frac{113-\sqrt{12289}}{-8}$$

Divide 113-\sqrt{12289} by -8.

$$x=\frac{\sqrt{12289}-113}{8}$$

The equation is now solved.

$$x=\frac{-\sqrt{12289}-113}{8} x=\frac{\sqrt{12289}-113}{8}$$

Steps for Completing the Square

Use the distributive property to multiply 4x by x+30.

$$7x=4x^{2}+120x+30$$

Subtract 4x^{2} from both sides.

$$7x-4x^{2}=120x+30$$

Subtract 120x from both sides.

$$7x-4x^{2}-120x=30$$

Combine 7x and -120x to get -113x.

$$-113x-4x^{2}=30$$

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.

$$-4x^{2}-113x=30$$

Divide both sides by -4.

$$\frac{-4x^{2}-113x}{-4}=\frac{30}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\left(-\frac{113}{-4}\right)x=\frac{30}{-4}$$

Divide -113 by -4.

$$x^{2}+\frac{113}{4}x=\frac{30}{-4}$$

Reduce the fraction \frac{30}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{113}{4}x=-\frac{15}{2}$$

Divide \frac{113}{4}, the coefficient of the x term, by 2 to get \frac{113}{8}. Then add the square of \frac{113}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{113}{4}x+\left(\frac{113}{8}\right)^{2}=-\frac{15}{2}+\left(\frac{113}{8}\right)^{2}$$

Square \frac{113}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{113}{4}x+\frac{12769}{64}=-\frac{15}{2}+\frac{12769}{64}$$

Add -\frac{15}{2} to \frac{12769}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{113}{4}x+\frac{12769}{64}=\frac{12289}{64}$$

Factor x^{2}+\frac{113}{4}x+\frac{12769}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{113}{8}\right)^{2}=\frac{12289}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{113}{8}\right)^{2}}=\sqrt{\frac{12289}{64}}$$

Simplify.

$$x+\frac{113}{8}=\frac{\sqrt{12289}}{8} x+\frac{113}{8}=-\frac{\sqrt{12289}}{8}$$

Subtract \frac{113}{8} from both sides of the equation.

$$x=\frac{\sqrt{12289}-113}{8} x=\frac{-\sqrt{12289}-113}{8}$$