Solve 21x^2+35x

$$21 x ^ { 2 } + 35 x$$

Factor

$$7x\left(3x+5\right)$$

Solution Steps

Factor out 7.

$$7\left(3x^{2}+5x\right)$$

Consider 3x^{2}+5x. Factor out x.

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

Rewrite the complete factored expression.

$$7x\left(3x+5\right)$$

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.

$$21x^{2}+35x=0$$

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.

$$x=\frac{-35±\sqrt{35^{2}}}{2\times 21}$$

Take the square root of 35^{2}.

$$x=\frac{-35±35}{2\times 21}$$

Multiply 2 times 21.

$$x=\frac{-35±35}{42}$$

Now solve the equation x=\frac{-35±35}{42} when ± is plus. Add -35 to 35.

$$x=\frac{0}{42}$$

Divide 0 by 42.

$$x=0$$

Now solve the equation x=\frac{-35±35}{42} when ± is minus. Subtract 35 from -35.

$$x=-\frac{70}{42}$$

Reduce the fraction \frac{-70}{42} to lowest terms by extracting and canceling out 14.

$$x=-\frac{5}{3}$$

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}{3} for x_{2}.

$$21x^{2}+35x=21x\left(x-\left(-\frac{5}{3}\right)\right)$$

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

$$21x^{2}+35x=21x\left(x+\frac{5}{3}\right)$$

Add \frac{5}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$21x^{2}+35x=21x\times \frac{3x+5}{3}$$

Cancel out 3, the greatest common factor in 21 and 3.

$$21x^{2}+35x=7x\left(3x+5\right)$$

Evaluate

$$7x\left(3x+5\right)$$