Solve 5x^2+24x-5

$$5 { x }^{ 2 } +24x-5$$

Factor

$$\left(5x-1\right)\left(x+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=24 ab=5\left(-5\right)=-25$$

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 -25.

$$-1,25 -5,5$$

Calculate the sum for each pair.

$$-1+25=24 -5+5=0$$

The solution is the pair that gives sum 24.

$$a=-1 b=25$$

Rewrite 5x^{2}+24x-5 as \left(5x^{2}-x\right)+\left(25x-5\right).

$$\left(5x^{2}-x\right)+\left(25x-5\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(5x-1\right)+5\left(5x-1\right)$$

Factor out common term 5x-1 by using distributive property.

$$\left(5x-1\right)\left(x+5\right)$$

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.

$$5x^{2}+24x-5=0$$

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.

$$x=\frac{-24±\sqrt{24^{2}-4\times 5\left(-5\right)}}{2\times 5}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 5\left(-5\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-24±\sqrt{576-20\left(-5\right)}}{2\times 5}$$

Multiply -20 times -5.

$$x=\frac{-24±\sqrt{576+100}}{2\times 5}$$

Add 576 to 100.

$$x=\frac{-24±\sqrt{676}}{2\times 5}$$

Take the square root of 676.

$$x=\frac{-24±26}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-24±26}{10}$$

Now solve the equation x=\frac{-24±26}{10} when ± is plus. Add -24 to 26.

$$x=\frac{2}{10}$$

Reduce the fraction \frac{2}{10} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{5}$$

Now solve the equation x=\frac{-24±26}{10} when ± is minus. Subtract 26 from -24.

$$x=-\frac{50}{10}$$

Divide -50 by 10.

$$x=-5$$

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

$$5x^{2}+24x-5=5\left(x-\frac{1}{5}\right)\left(x-\left(-5\right)\right)$$

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

$$5x^{2}+24x-5=5\left(x-\frac{1}{5}\right)\left(x+5\right)$$

Subtract \frac{1}{5} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$5x^{2}+24x-5=5\times \frac{5x-1}{5}\left(x+5\right)$$

Cancel out 5, the greatest common factor in 5 and 5.

$$5x^{2}+24x-5=\left(5x-1\right)\left(x+5\right)$$

Evaluate

$$\left(5x-1\right)\left(x+5\right)$$