Solve 4a^2+7a

$$4 a ^ { 2 } + 7 a$$

Factor

$$a\left(4a+7\right)$$

Solution Steps

Factor out a.

$$a\left(4a+7\right)$$

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.

$$4a^{2}+7a=0$$

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.

$$a=\frac{-7±\sqrt{7^{2}}}{2\times 4}$$

Take the square root of 7^{2}.

$$a=\frac{-7±7}{2\times 4}$$

Multiply 2 times 4.

$$a=\frac{-7±7}{8}$$

Now solve the equation a=\frac{-7±7}{8} when ± is plus. Add -7 to 7.

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

Divide 0 by 8.

$$a=0$$

Now solve the equation a=\frac{-7±7}{8} when ± is minus. Subtract 7 from -7.

$$a=-\frac{14}{8}$$

Reduce the fraction \frac{-14}{8} to lowest terms by extracting and canceling out 2.

$$a=-\frac{7}{4}$$

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{7}{4} for x_{2}.

$$4a^{2}+7a=4a\left(a-\left(-\frac{7}{4}\right)\right)$$

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

$$4a^{2}+7a=4a\left(a+\frac{7}{4}\right)$$

Add \frac{7}{4} to a by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$4a^{2}+7a=4a\times \frac{4a+7}{4}$$

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

$$4a^{2}+7a=a\left(4a+7\right)$$

Evaluate

$$a\left(4a+7\right)$$