Solve 24x^2+21x

$$24 x ^ { 2 } + 21 x$$

Factor

$$3x\left(8x+7\right)$$

Solution Steps

Factor out 3.

$$3\left(8x^{2}+7x\right)$$

Consider 8x^{2}+7x. Factor out x.

$$x\left(8x+7\right)$$

Rewrite the complete factored expression.

$$3x\left(8x+7\right)$$

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.

$$24x^{2}+21x=0$$

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.

$$x=\frac{-21±\sqrt{21^{2}}}{2\times 24}$$

Take the square root of 21^{2}.

$$x=\frac{-21±21}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{-21±21}{48}$$

Now solve the equation x=\frac{-21±21}{48} when ± is plus. Add -21 to 21.

$$x=\frac{0}{48}$$

Divide 0 by 48.

$$x=0$$

Now solve the equation x=\frac{-21±21}{48} when ± is minus. Subtract 21 from -21.

$$x=-\frac{42}{48}$$

Reduce the fraction \frac{-42}{48} to lowest terms by extracting and canceling out 6.

$$x=-\frac{7}{8}$$

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}{8} for x_{2}.

$$24x^{2}+21x=24x\left(x-\left(-\frac{7}{8}\right)\right)$$

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

$$24x^{2}+21x=24x\left(x+\frac{7}{8}\right)$$

Add \frac{7}{8} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$24x^{2}+21x=24x\times \frac{8x+7}{8}$$

Cancel out 8, the greatest common factor in 24 and 8.

$$24x^{2}+21x=3x\left(8x+7\right)$$

Evaluate

$$3x\left(8x+7\right)$$