Solve (6-x)(52+x)=25

$$(6-x)(52+x)=25$$

Solve for x

$$x=4\sqrt{51}-23\approx 5.565713714$$

$$x=-4\sqrt{51}-23\approx -51.565713714$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6-x by 52+x and combine like terms.

$$312-46x-x^{2}=25$$

Subtract 25 from both sides.

$$312-46x-x^{2}-25=0$$

Subtract 25 from 312 to get 287.

$$287-46x-x^{2}=0$$

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.

$$-x^{2}-46x+287=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -46 for b, and 287 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-46\right)±\sqrt{\left(-46\right)^{2}-4\left(-1\right)\times 287}}{2\left(-1\right)}$$

Square -46.

$$x=\frac{-\left(-46\right)±\sqrt{2116-4\left(-1\right)\times 287}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-46\right)±\sqrt{2116+4\times 287}}{2\left(-1\right)}$$

Multiply 4 times 287.

$$x=\frac{-\left(-46\right)±\sqrt{2116+1148}}{2\left(-1\right)}$$

Add 2116 to 1148.

$$x=\frac{-\left(-46\right)±\sqrt{3264}}{2\left(-1\right)}$$

Take the square root of 3264.

$$x=\frac{-\left(-46\right)±8\sqrt{51}}{2\left(-1\right)}$$

The opposite of -46 is 46.

$$x=\frac{46±8\sqrt{51}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{46±8\sqrt{51}}{-2}$$

Now solve the equation x=\frac{46±8\sqrt{51}}{-2} when ± is plus. Add 46 to 8\sqrt{51}.

$$x=\frac{8\sqrt{51}+46}{-2}$$

Divide 46+8\sqrt{51} by -2.

$$x=-4\sqrt{51}-23$$

Now solve the equation x=\frac{46±8\sqrt{51}}{-2} when ± is minus. Subtract 8\sqrt{51} from 46.

$$x=\frac{46-8\sqrt{51}}{-2}$$

Divide 46-8\sqrt{51} by -2.

$$x=4\sqrt{51}-23$$

The equation is now solved.

$$x=-4\sqrt{51}-23 x=4\sqrt{51}-23$$

Steps for Completing the Square

Use the distributive property to multiply 6-x by 52+x and combine like terms.

$$312-46x-x^{2}=25$$

Subtract 312 from both sides.

$$-46x-x^{2}=25-312$$

Subtract 312 from 25 to get -287.

$$-46x-x^{2}=-287$$

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.

$$-x^{2}-46x=-287$$

Divide both sides by -1.

$$\frac{-x^{2}-46x}{-1}=-\frac{287}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{46}{-1}\right)x=-\frac{287}{-1}$$

Divide -46 by -1.

$$x^{2}+46x=-\frac{287}{-1}$$

Divide -287 by -1.

$$x^{2}+46x=287$$

Divide 46, the coefficient of the x term, by 2 to get 23. Then add the square of 23 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+46x+23^{2}=287+23^{2}$$

Square 23.

$$x^{2}+46x+529=287+529$$

Add 287 to 529.

$$x^{2}+46x+529=816$$

Factor x^{2}+46x+529. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+23\right)^{2}=816$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+23\right)^{2}}=\sqrt{816}$$

Simplify.

$$x+23=4\sqrt{51} x+23=-4\sqrt{51}$$

Subtract 23 from both sides of the equation.

$$x=4\sqrt{51}-23 x=-4\sqrt{51}-23$$