Solve 5a^2-21a=20

$$5 a ^ { 2 } - 21 a = 20$$

Solve for a

$$a=-\frac{4}{5}=-0.8$$

$$a=5$$

Steps Using Factoring By Grouping

Subtract 20 from both sides.

$$5a^{2}-21a-20=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 5a^{2}+aa+ba-20. To find a and b, set up a system to be solved.

$$a+b=-21 ab=5\left(-20\right)=-100$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -100.

$$1,-100 2,-50 4,-25 5,-20 10,-10$$

Calculate the sum for each pair.

$$1-100=-99 2-50=-48 4-25=-21 5-20=-15 10-10=0$$

The solution is the pair that gives sum -21.

$$a=-25 b=4$$

Rewrite 5a^{2}-21a-20 as \left(5a^{2}-25a\right)+\left(4a-20\right).

$$\left(5a^{2}-25a\right)+\left(4a-20\right)$$

Factor out 5a in the first and 4 in the second group.

$$5a\left(a-5\right)+4\left(a-5\right)$$

Factor out common term a-5 by using distributive property.

$$\left(a-5\right)\left(5a+4\right)$$

To find equation solutions, solve a-5=0 and 5a+4=0.

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

Steps Using the Quadratic Formula

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.

$$5a^{2}-21a=20$$

Subtract 20 from both sides of the equation.

$$5a^{2}-21a-20=20-20$$

Subtracting 20 from itself leaves 0.

$$5a^{2}-21a-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -21 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 5\left(-20\right)}}{2\times 5}$$

Square -21.

$$a=\frac{-\left(-21\right)±\sqrt{441-4\times 5\left(-20\right)}}{2\times 5}$$

Multiply -4 times 5.

$$a=\frac{-\left(-21\right)±\sqrt{441-20\left(-20\right)}}{2\times 5}$$

Multiply -20 times -20.

$$a=\frac{-\left(-21\right)±\sqrt{441+400}}{2\times 5}$$

Add 441 to 400.

$$a=\frac{-\left(-21\right)±\sqrt{841}}{2\times 5}$$

Take the square root of 841.

$$a=\frac{-\left(-21\right)±29}{2\times 5}$$

The opposite of -21 is 21.

$$a=\frac{21±29}{2\times 5}$$

Multiply 2 times 5.

$$a=\frac{21±29}{10}$$

Now solve the equation a=\frac{21±29}{10} when ± is plus. Add 21 to 29.

$$a=\frac{50}{10}$$

Divide 50 by 10.

$$a=5$$

Now solve the equation a=\frac{21±29}{10} when ± is minus. Subtract 29 from 21.

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

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

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

The equation is now solved.

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

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$5a^{2}-21a=20$$

Divide both sides by 5.

$$\frac{5a^{2}-21a}{5}=\frac{20}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$a^{2}-\frac{21}{5}a=\frac{20}{5}$$

Divide 20 by 5.

$$a^{2}-\frac{21}{5}a=4$$

Divide -\frac{21}{5}, the coefficient of the x term, by 2 to get -\frac{21}{10}. Then add the square of -\frac{21}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{21}{5}a+\left(-\frac{21}{10}\right)^{2}=4+\left(-\frac{21}{10}\right)^{2}$$

Square -\frac{21}{10} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{21}{5}a+\frac{441}{100}=4+\frac{441}{100}$$

Add 4 to \frac{441}{100}.

$$a^{2}-\frac{21}{5}a+\frac{441}{100}=\frac{841}{100}$$

Factor a^{2}-\frac{21}{5}a+\frac{441}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{21}{10}\right)^{2}=\frac{841}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{21}{10}\right)^{2}}=\sqrt{\frac{841}{100}}$$

Simplify.

$$a-\frac{21}{10}=\frac{29}{10} a-\frac{21}{10}=-\frac{29}{10}$$

Add \frac{21}{10} to both sides of the equation.

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