Solve x^2=64(4,25-x)

$$x ^ { 2 } = 64 ( 4,25 - x )$$

Solve for x

$$x=-68$$

$$x=4$$

Steps Using Factoring

Use the distributive property to multiply 64 by 4,25-x.

$$x^{2}=272-64x$$

Subtract 272 from both sides.

$$x^{2}-272=-64x$$

Add 64x to both sides.

$$x^{2}-272+64x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}+64x-272=0$$

To solve the equation, factor x^{2}+64x-272 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=64 ab=-272$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -272.

$$-1;272 -2;136 -4;68 -8;34 -16;17$$

Calculate the sum for each pair.

$$-1+272=271 -2+136=134 -4+68=64 -8+34=26 -16+17=1$$

The solution is the pair that gives sum 64.

$$a=-4 b=68$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-4\right)\left(x+68\right)$$

To find equation solutions, solve x-4=0 and x+68=0.

$$x=4 x=-68$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 64 by 4,25-x.

$$x^{2}=272-64x$$

Subtract 272 from both sides.

$$x^{2}-272=-64x$$

Add 64x to both sides.

$$x^{2}-272+64x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}+64x-272=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-272. To find a and b, set up a system to be solved.

$$a+b=64 ab=1\left(-272\right)=-272$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -272.

$$-1;272 -2;136 -4;68 -8;34 -16;17$$

Calculate the sum for each pair.

$$-1+272=271 -2+136=134 -4+68=64 -8+34=26 -16+17=1$$

The solution is the pair that gives sum 64.

$$a=-4 b=68$$

Rewrite x^{2}+64x-272 as \left(x^{2}-4x\right)+\left(68x-272\right).

$$\left(x^{2}-4x\right)+\left(68x-272\right)$$

Factor out x in the first and 68 in the second group.

$$x\left(x-4\right)+68\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(x+68\right)$$

To find equation solutions, solve x-4=0 and x+68=0.

$$x=4 x=-68$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 64 by 4,25-x.

$$x^{2}=272-64x$$

Subtract 272 from both sides.

$$x^{2}-272=-64x$$

Add 64x to both sides.

$$x^{2}-272+64x=0$$

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.

$$x^{2}+64x-272=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 64 for b, and -272 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-64±\sqrt{64^{2}-4\left(-272\right)}}{2}$$

Square 64.

$$x=\frac{-64±\sqrt{4096-4\left(-272\right)}}{2}$$

Multiply -4 times -272.

$$x=\frac{-64±\sqrt{4096+1088}}{2}$$

Add 4096 to 1088.

$$x=\frac{-64±\sqrt{5184}}{2}$$

Take the square root of 5184.

$$x=\frac{-64±72}{2}$$

Now solve the equation x=\frac{-64±72}{2} when ± is plus. Add -64 to 72.

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

Now solve the equation x=\frac{-64±72}{2} when ± is minus. Subtract 72 from -64.

$$x=-\frac{136}{2}$$

Divide -136 by 2.

$$x=-68$$

The equation is now solved.

$$x=4 x=-68$$

Steps for Completing the Square

Use the distributive property to multiply 64 by 4,25-x.

$$x^{2}=272-64x$$

Add 64x to both sides.

$$x^{2}+64x=272$$

Divide 64, the coefficient of the x term, by 2 to get 32. Then add the square of 32 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+64x+32^{2}=272+32^{2}$$

Square 32.

$$x^{2}+64x+1024=272+1024$$

Add 272 to 1024.

$$x^{2}+64x+1024=1296$$

Factor x^{2}+64x+1024. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+32\right)^{2}=1296$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+32\right)^{2}}=\sqrt{1296}$$

Simplify.

$$x+32=36 x+32=-36$$

Subtract 32 from both sides of the equation.

$$x=4 x=-68$$