Solve 4x^2-8x-60

$$4 { x }^{ 2 } -8x-60$$

Factor

$$4\left(x-5\right)\left(x+3\right)$$

Solution Steps

Factor out 4.

$$4\left(x^{2}-2x-15\right)$$

Consider x^{2}-2x-15. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-15\right)=-15$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -15.

$$1,-15 3,-5$$

Calculate the sum for each pair.

$$1-15=-14 3-5=-2$$

The solution is the pair that gives sum -2.

$$a=-5 b=3$$

Rewrite x^{2}-2x-15 as \left(x^{2}-5x\right)+\left(3x-15\right).

$$\left(x^{2}-5x\right)+\left(3x-15\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(x-5\right)+3\left(x-5\right)$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(x+3\right)$$

Rewrite the complete factored expression.

$$4\left(x-5\right)\left(x+3\right)$$

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.

$$4x^{2}-8x-60=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 4\left(-60\right)}}{2\times 4}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 4\left(-60\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-8\right)±\sqrt{64-16\left(-60\right)}}{2\times 4}$$

Multiply -16 times -60.

$$x=\frac{-\left(-8\right)±\sqrt{64+960}}{2\times 4}$$

Add 64 to 960.

$$x=\frac{-\left(-8\right)±\sqrt{1024}}{2\times 4}$$

Take the square root of 1024.

$$x=\frac{-\left(-8\right)±32}{2\times 4}$$

The opposite of -8 is 8.

$$x=\frac{8±32}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{8±32}{8}$$

Now solve the equation x=\frac{8±32}{8} when ± is plus. Add 8 to 32.

$$x=\frac{40}{8}$$

Divide 40 by 8.

$$x=5$$

Now solve the equation x=\frac{8±32}{8} when ± is minus. Subtract 32 from 8.

$$x=-\frac{24}{8}$$

Divide -24 by 8.

$$x=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and -3 for x_{2}.

$$4x^{2}-8x-60=4\left(x-5\right)\left(x-\left(-3\right)\right)$$

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

$$4x^{2}-8x-60=4\left(x-5\right)\left(x+3\right)$$

Evaluate

$$4\left(x-5\right)\left(x+3\right)$$