Solve x^2-60=-28x

$${ x }^{ 2 } -60=-28x$$

Solve for x

$$x=-30$$

$$x=2$$

Steps Using Factoring

Add 28x to both sides.

$$x^{2}-60+28x=0$$

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

$$x^{2}+28x-60=0$$

To solve the equation, factor x^{2}+28x-60 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.

$$a+b=28 ab=-60$$

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 -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Calculate the sum for each pair.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

The solution is the pair that gives sum 28.

$$a=-2 b=30$$

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

$$\left(x-2\right)\left(x+30\right)$$

To find equation solutions, solve x-2=0 and x+30=0.

$$x=2 x=-30$$

Steps Using Factoring By Grouping

Add 28x to both sides.

$$x^{2}-60+28x=0$$

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

$$x^{2}+28x-60=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-60. To find a and b, set up a system to be solved.

$$a+b=28 ab=1\left(-60\right)=-60$$

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 -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Calculate the sum for each pair.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

The solution is the pair that gives sum 28.

$$a=-2 b=30$$

Rewrite x^{2}+28x-60 as \left(x^{2}-2x\right)+\left(30x-60\right).

$$\left(x^{2}-2x\right)+\left(30x-60\right)$$

Factor out x in the first and 30 in the second group.

$$x\left(x-2\right)+30\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(x+30\right)$$

To find equation solutions, solve x-2=0 and x+30=0.

$$x=2 x=-30$$

Steps Using the Quadratic Formula

Add 28x to both sides.

$$x^{2}-60+28x=0$$

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.

$$x^{2}+28x-60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 28 for b, and -60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-60\right)}}{2}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-60\right)}}{2}$$

Multiply -4 times -60.

$$x=\frac{-28±\sqrt{784+240}}{2}$$

Add 784 to 240.

$$x=\frac{-28±\sqrt{1024}}{2}$$

Take the square root of 1024.

$$x=\frac{-28±32}{2}$$

Now solve the equation x=\frac{-28±32}{2} when ± is plus. Add -28 to 32.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{-28±32}{2} when ± is minus. Subtract 32 from -28.

$$x=-\frac{60}{2}$$

Divide -60 by 2.

$$x=-30$$

The equation is now solved.

$$x=2 x=-30$$

Steps for Completing the Square

Add 28x to both sides.

$$x^{2}-60+28x=0$$

Add 60 to both sides. Anything plus zero gives itself.

$$x^{2}+28x=60$$

Divide 28, the coefficient of the x term, by 2 to get 14. Then add the square of 14 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+28x+14^{2}=60+14^{2}$$

Square 14.

$$x^{2}+28x+196=60+196$$

Add 60 to 196.

$$x^{2}+28x+196=256$$

Factor x^{2}+28x+196. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+14\right)^{2}=256$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+14\right)^{2}}=\sqrt{256}$$

Simplify.

$$x+14=16 x+14=-16$$

Subtract 14 from both sides of the equation.

$$x=2 x=-30$$