Solve 3*(x-12)+23=5x(25-x)

$$3 \times (x-12)+23=5x(25-x)$$

Solve for x

$$x = \frac{\sqrt{3786} + 61}{5} \approx 24.50609605$$

$$x=\frac{61-\sqrt{3786}}{5}\approx -0.10609605$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3 by x-12.

$$3x-36+23=5x\left(25-x\right)$$

Add -36 and 23 to get -13.

$$3x-13=5x\left(25-x\right)$$

Use the distributive property to multiply 5x by 25-x.

$$3x-13=125x-5x^{2}$$

Subtract 125x from both sides.

$$3x-13-125x=-5x^{2}$$

Combine 3x and -125x to get -122x.

$$-122x-13=-5x^{2}$$

Add 5x^{2} to both sides.

$$-122x-13+5x^{2}=0$$

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.

$$5x^{2}-122x-13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -122 for b, and -13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-122\right)±\sqrt{\left(-122\right)^{2}-4\times 5\left(-13\right)}}{2\times 5}$$

Square -122.

$$x=\frac{-\left(-122\right)±\sqrt{14884-4\times 5\left(-13\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-122\right)±\sqrt{14884-20\left(-13\right)}}{2\times 5}$$

Multiply -20 times -13.

$$x=\frac{-\left(-122\right)±\sqrt{14884+260}}{2\times 5}$$

Add 14884 to 260.

$$x=\frac{-\left(-122\right)±\sqrt{15144}}{2\times 5}$$

Take the square root of 15144.

$$x=\frac{-\left(-122\right)±2\sqrt{3786}}{2\times 5}$$

The opposite of -122 is 122.

$$x=\frac{122±2\sqrt{3786}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{122±2\sqrt{3786}}{10}$$

Now solve the equation x=\frac{122±2\sqrt{3786}}{10} when ± is plus. Add 122 to 2\sqrt{3786}.

$$x=\frac{2\sqrt{3786}+122}{10}$$

Divide 122+2\sqrt{3786} by 10.

$$x=\frac{\sqrt{3786}+61}{5}$$

Now solve the equation x=\frac{122±2\sqrt{3786}}{10} when ± is minus. Subtract 2\sqrt{3786} from 122.

$$x=\frac{122-2\sqrt{3786}}{10}$$

Divide 122-2\sqrt{3786} by 10.

$$x=\frac{61-\sqrt{3786}}{5}$$

The equation is now solved.

$$x=\frac{\sqrt{3786}+61}{5} x=\frac{61-\sqrt{3786}}{5}$$

Steps for Completing the Square

Use the distributive property to multiply 3 by x-12.

$$3x-36+23=5x\left(25-x\right)$$

Add -36 and 23 to get -13.

$$3x-13=5x\left(25-x\right)$$

Use the distributive property to multiply 5x by 25-x.

$$3x-13=125x-5x^{2}$$

Subtract 125x from both sides.

$$3x-13-125x=-5x^{2}$$

Combine 3x and -125x to get -122x.

$$-122x-13=-5x^{2}$$

Add 5x^{2} to both sides.

$$-122x-13+5x^{2}=0$$

Add 13 to both sides. Anything plus zero gives itself.

$$-122x+5x^{2}=13$$

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.

$$5x^{2}-122x=13$$

Divide both sides by 5.

$$\frac{5x^{2}-122x}{5}=\frac{13}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{122}{5}x=\frac{13}{5}$$

Divide -\frac{122}{5}, the coefficient of the x term, by 2 to get -\frac{61}{5}. Then add the square of -\frac{61}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{122}{5}x+\left(-\frac{61}{5}\right)^{2}=\frac{13}{5}+\left(-\frac{61}{5}\right)^{2}$$

Square -\frac{61}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{122}{5}x+\frac{3721}{25}=\frac{13}{5}+\frac{3721}{25}$$

Add \frac{13}{5} to \frac{3721}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{122}{5}x+\frac{3721}{25}=\frac{3786}{25}$$

Factor x^{2}-\frac{122}{5}x+\frac{3721}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{61}{5}\right)^{2}=\frac{3786}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{61}{5}\right)^{2}}=\sqrt{\frac{3786}{25}}$$

Simplify.

$$x-\frac{61}{5}=\frac{\sqrt{3786}}{5} x-\frac{61}{5}=-\frac{\sqrt{3786}}{5}$$

Add \frac{61}{5} to both sides of the equation.

$$x=\frac{\sqrt{3786}+61}{5} x=\frac{61-\sqrt{3786}}{5}$$