Solve (76+x)*x=152*76

$$(76+x) \times x=152 \times 76$$

Solve for x

$$x=-152$$

$$x=76$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 76+x by x.

$$76x+x^{2}=152\times 76$$

Multiply 152 and 76 to get 11552.

$$76x+x^{2}=11552$$

Subtract 11552 from both sides.

$$76x+x^{2}-11552=0$$

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.

$$x^{2}+76x-11552=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 76 for b, and -11552 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-76±\sqrt{76^{2}-4\left(-11552\right)}}{2}$$

Square 76.

$$x=\frac{-76±\sqrt{5776-4\left(-11552\right)}}{2}$$

Multiply -4 times -11552.

$$x=\frac{-76±\sqrt{5776+46208}}{2}$$

Add 5776 to 46208.

$$x=\frac{-76±\sqrt{51984}}{2}$$

Take the square root of 51984.

$$x=\frac{-76±228}{2}$$

Now solve the equation x=\frac{-76±228}{2} when ± is plus. Add -76 to 228.

$$x=\frac{152}{2}$$

Divide 152 by 2.

$$x=76$$

Now solve the equation x=\frac{-76±228}{2} when ± is minus. Subtract 228 from -76.

$$x=-\frac{304}{2}$$

Divide -304 by 2.

$$x=-152$$

The equation is now solved.

$$x=76 x=-152$$

Steps for Completing the Square

Use the distributive property to multiply 76+x by x.

$$76x+x^{2}=152\times 76$$

Multiply 152 and 76 to get 11552.

$$76x+x^{2}=11552$$

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.

$$x^{2}+76x=11552$$

Divide 76, the coefficient of the x term, by 2 to get 38. Then add the square of 38 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+76x+38^{2}=11552+38^{2}$$

Square 38.

$$x^{2}+76x+1444=11552+1444$$

Add 11552 to 1444.

$$x^{2}+76x+1444=12996$$

Factor x^{2}+76x+1444. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+38\right)^{2}=12996$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+38\right)^{2}}=\sqrt{12996}$$

Simplify.

$$x+38=114 x+38=-114$$

Subtract 38 from both sides of the equation.

$$x=76 x=-152$$