Solve 24x=31-(4-3x)^2

$$24 x = 31 - ( 4 - 3 x ) ^ { 2 }$$

Solve for x

$$x = \frac{\sqrt{15}}{3} \approx 1.290994449$$

$$x = -\frac{\sqrt{15}}{3} \approx -1.290994449$$

Steps by Finding Square Root

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-3x\right)^{2}.

$$24x=31-\left(16-24x+9x^{2}\right)$$

To find the opposite of 16-24x+9x^{2}, find the opposite of each term.

$$24x=31-16+24x-9x^{2}$$

Subtract 16 from 31 to get 15.

$$24x=15+24x-9x^{2}$$

Subtract 24x from both sides.

$$24x-24x=15-9x^{2}$$

Combine 24x and -24x to get 0.

$$0=15-9x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$15-9x^{2}=0$$

Subtract 15 from both sides. Anything subtracted from zero gives its negation.

$$-9x^{2}=-15$$

Divide both sides by -9.

$$x^{2}=\frac{-15}{-9}$$

Reduce the fraction \frac{-15}{-9} to lowest terms by extracting and canceling out -3.

$$x^{2}=\frac{5}{3}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{15}}{3} x=-\frac{\sqrt{15}}{3}$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-3x\right)^{2}.

$$24x=31-\left(16-24x+9x^{2}\right)$$

To find the opposite of 16-24x+9x^{2}, find the opposite of each term.

$$24x=31-16+24x-9x^{2}$$

Subtract 16 from 31 to get 15.

$$24x=15+24x-9x^{2}$$

Subtract 15 from both sides.

$$24x-15=24x-9x^{2}$$

Subtract 24x from both sides.

$$24x-15-24x=-9x^{2}$$

Combine 24x and -24x to get 0.

$$-15=-9x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$-9x^{2}=-15$$

Add 15 to both sides.

$$-9x^{2}+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 0 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-9\right)\times 15}}{2\left(-9\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-9\right)\times 15}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{0±\sqrt{36\times 15}}{2\left(-9\right)}$$

Multiply 36 times 15.

$$x=\frac{0±\sqrt{540}}{2\left(-9\right)}$$

Take the square root of 540.

$$x=\frac{0±6\sqrt{15}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{0±6\sqrt{15}}{-18}$$

Now solve the equation x=\frac{0±6\sqrt{15}}{-18} when ± is plus.

$$x=-\frac{\sqrt{15}}{3}$$

Now solve the equation x=\frac{0±6\sqrt{15}}{-18} when ± is minus.

$$x=\frac{\sqrt{15}}{3}$$

The equation is now solved.

$$x=-\frac{\sqrt{15}}{3} x=\frac{\sqrt{15}}{3}$$