Solve f(x)=4x^2+5x

$$f ( x ) = 4 x ^ { 2 } + 5 x$$

Factor

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

Solution Steps

Factor out x.

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

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.

$$4x^{2}+5x=0$$

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.

$$x=\frac{-5±\sqrt{5^{2}}}{2\times 4}$$

Take the square root of 5^{2}.

$$x=\frac{-5±5}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-5±5}{8}$$

Now solve the equation x=\frac{-5±5}{8} when ± is plus. Add -5 to 5.

$$x=\frac{0}{8}$$

Divide 0 by 8.

$$x=0$$

Now solve the equation x=\frac{-5±5}{8} when ± is minus. Subtract 5 from -5.

$$x=-\frac{10}{8}$$

Reduce the fraction \frac{-10}{8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{5}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 0 for x_{1} and -\frac{5}{4} for x_{2}.

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

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

$$4x^{2}+5x=4x\left(x+\frac{5}{4}\right)$$

Add \frac{5}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$4x^{2}+5x=4x\times \frac{4x+5}{4}$$

Cancel out 4, the greatest common factor in 4 and 4.

$$4x^{2}+5x=x\left(4x+5\right)$$

Evaluate

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