Solve (x-15)(58-2x)=90

$$(x-15)(58-2x)=90$$

Solve for x

$$x=20$$

$$x=24$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-15 by 58-2x and combine like terms.

$$88x-2x^{2}-870=90$$

Subtract 90 from both sides.

$$88x-2x^{2}-870-90=0$$

Subtract 90 from -870 to get -960.

$$88x-2x^{2}-960=0$$

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.

$$-2x^{2}+88x-960=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 88 for b, and -960 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-88±\sqrt{88^{2}-4\left(-2\right)\left(-960\right)}}{2\left(-2\right)}$$

Square 88.

$$x=\frac{-88±\sqrt{7744-4\left(-2\right)\left(-960\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-88±\sqrt{7744+8\left(-960\right)}}{2\left(-2\right)}$$

Multiply 8 times -960.

$$x=\frac{-88±\sqrt{7744-7680}}{2\left(-2\right)}$$

Add 7744 to -7680.

$$x=\frac{-88±\sqrt{64}}{2\left(-2\right)}$$

Take the square root of 64.

$$x=\frac{-88±8}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-88±8}{-4}$$

Now solve the equation x=\frac{-88±8}{-4} when ± is plus. Add -88 to 8.

$$x=-\frac{80}{-4}$$

Divide -80 by -4.

$$x=20$$

Now solve the equation x=\frac{-88±8}{-4} when ± is minus. Subtract 8 from -88.

$$x=-\frac{96}{-4}$$

Divide -96 by -4.

$$x=24$$

The equation is now solved.

$$x=20 x=24$$

Steps for Completing the Square

Use the distributive property to multiply x-15 by 58-2x and combine like terms.

$$88x-2x^{2}-870=90$$

Add 870 to both sides.

$$88x-2x^{2}=90+870$$

Add 90 and 870 to get 960.

$$88x-2x^{2}=960$$

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.

$$-2x^{2}+88x=960$$

Divide both sides by -2.

$$\frac{-2x^{2}+88x}{-2}=\frac{960}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{88}{-2}x=\frac{960}{-2}$$

Divide 88 by -2.

$$x^{2}-44x=\frac{960}{-2}$$

Divide 960 by -2.

$$x^{2}-44x=-480$$

Divide -44, the coefficient of the x term, by 2 to get -22. Then add the square of -22 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-44x+\left(-22\right)^{2}=-480+\left(-22\right)^{2}$$

Square -22.

$$x^{2}-44x+484=-480+484$$

Add -480 to 484.

$$x^{2}-44x+484=4$$

Factor x^{2}-44x+484. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-22\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-22\right)^{2}}=\sqrt{4}$$

Simplify.

$$x-22=2 x-22=-2$$

Add 22 to both sides of the equation.

$$x=24 x=20$$