Solve -13x^2+130x+312

$$-13 { x }^{ 2 } +130x+312$$

Factor

$$-13\left(x-12\right)\left(x+2\right)$$

Solution Steps

Factor out 13.

$$13\left(-x^{2}+10x+24\right)$$

Consider -x^{2}+10x+24. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+24. To find a and b, set up a system to be solved.

$$a+b=10 ab=-24=-24$$

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 -24.

$$-1,24 -2,12 -3,8 -4,6$$

Calculate the sum for each pair.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

The solution is the pair that gives sum 10.

$$a=12 b=-2$$

Rewrite -x^{2}+10x+24 as \left(-x^{2}+12x\right)+\left(-2x+24\right).

$$\left(-x^{2}+12x\right)+\left(-2x+24\right)$$

Factor out -x in the first and -2 in the second group.

$$-x\left(x-12\right)-2\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(-x-2\right)$$

Rewrite the complete factored expression.

$$13\left(x-12\right)\left(-x-2\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-13x^{2}+130x+312=0$$

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.

$$x=\frac{-130±\sqrt{130^{2}-4\left(-13\right)\times 312}}{2\left(-13\right)}$$

Square 130.

$$x=\frac{-130±\sqrt{16900-4\left(-13\right)\times 312}}{2\left(-13\right)}$$

Multiply -4 times -13.

$$x=\frac{-130±\sqrt{16900+52\times 312}}{2\left(-13\right)}$$

Multiply 52 times 312.

$$x=\frac{-130±\sqrt{16900+16224}}{2\left(-13\right)}$$

Add 16900 to 16224.

$$x=\frac{-130±\sqrt{33124}}{2\left(-13\right)}$$

Take the square root of 33124.

$$x=\frac{-130±182}{2\left(-13\right)}$$

Multiply 2 times -13.

$$x=\frac{-130±182}{-26}$$

Now solve the equation x=\frac{-130±182}{-26} when ± is plus. Add -130 to 182.

$$x=\frac{52}{-26}$$

Divide 52 by -26.

$$x=-2$$

Now solve the equation x=\frac{-130±182}{-26} when ± is minus. Subtract 182 from -130.

$$x=-\frac{312}{-26}$$

Divide -312 by -26.

$$x=12$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2 for x_{1} and 12 for x_{2}.

$$-13x^{2}+130x+312=-13\left(x-\left(-2\right)\right)\left(x-12\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-13x^{2}+130x+312=-13\left(x+2\right)\left(x-12\right)$$

Evaluate

$$-13\left(x-12\right)\left(x+2\right)$$