Solve -4a^2-1/4a=17/8

$$\frac{ -4 { a }^{ 2 } -1 }{ 4a } = \frac{ 17 }{ 8 }$$

Solve for a

$$a=-2$$

$$a=-\frac{1}{8}=-0.125$$

Steps Using Factoring By Grouping

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 8a, the least common multiple of 4a,8.

$$2\left(-4a^{2}-1\right)=17a$$

Use the distributive property to multiply 2 by -4a^{2}-1.

$$-8a^{2}-2=17a$$

Subtract 17a from both sides.

$$-8a^{2}-2-17a=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-8a^{2}-17a-2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -8a^{2}+aa+ba-2. To find a and b, set up a system to be solved.

$$a+b=-17 ab=-8\left(-2\right)=16$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 16.

$$-1,-16 -2,-8 -4,-4$$

Calculate the sum for each pair.

$$-1-16=-17 -2-8=-10 -4-4=-8$$

The solution is the pair that gives sum -17.

$$a=-1 b=-16$$

Rewrite -8a^{2}-17a-2 as \left(-8a^{2}-a\right)+\left(-16a-2\right).

$$\left(-8a^{2}-a\right)+\left(-16a-2\right)$$

Factor out -a in the first and -2 in the second group.

$$-a\left(8a+1\right)-2\left(8a+1\right)$$

Factor out common term 8a+1 by using distributive property.

$$\left(8a+1\right)\left(-a-2\right)$$

To find equation solutions, solve 8a+1=0 and -a-2=0.

$$a=-\frac{1}{8} a=-2$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 8a, the least common multiple of 4a,8.

$$2\left(-4a^{2}-1\right)=17a$$

Use the distributive property to multiply 2 by -4a^{2}-1.

$$-8a^{2}-2=17a$$

Subtract 17a from both sides.

$$-8a^{2}-2-17a=0$$

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.

$$-8a^{2}-17a-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, -17 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\left(-8\right)\left(-2\right)}}{2\left(-8\right)}$$

Square -17.

$$a=\frac{-\left(-17\right)±\sqrt{289-4\left(-8\right)\left(-2\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$a=\frac{-\left(-17\right)±\sqrt{289+32\left(-2\right)}}{2\left(-8\right)}$$

Multiply 32 times -2.

$$a=\frac{-\left(-17\right)±\sqrt{289-64}}{2\left(-8\right)}$$

Add 289 to -64.

$$a=\frac{-\left(-17\right)±\sqrt{225}}{2\left(-8\right)}$$

Take the square root of 225.

$$a=\frac{-\left(-17\right)±15}{2\left(-8\right)}$$

The opposite of -17 is 17.

$$a=\frac{17±15}{2\left(-8\right)}$$

Multiply 2 times -8.

$$a=\frac{17±15}{-16}$$

Now solve the equation a=\frac{17±15}{-16} when ± is plus. Add 17 to 15.

$$a=\frac{32}{-16}$$

Divide 32 by -16.

$$a=-2$$

Now solve the equation a=\frac{17±15}{-16} when ± is minus. Subtract 15 from 17.

$$a=\frac{2}{-16}$$

Reduce the fraction \frac{2}{-16} to lowest terms by extracting and canceling out 2.

$$a=-\frac{1}{8}$$

The equation is now solved.

$$a=-2 a=-\frac{1}{8}$$

Steps for Completing the Square

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 8a, the least common multiple of 4a,8.

$$2\left(-4a^{2}-1\right)=17a$$

Use the distributive property to multiply 2 by -4a^{2}-1.

$$-8a^{2}-2=17a$$

Subtract 17a from both sides.

$$-8a^{2}-2-17a=0$$

Add 2 to both sides. Anything plus zero gives itself.

$$-8a^{2}-17a=2$$

Divide both sides by -8.

$$\frac{-8a^{2}-17a}{-8}=\frac{2}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$a^{2}+\left(-\frac{17}{-8}\right)a=\frac{2}{-8}$$

Divide -17 by -8.

$$a^{2}+\frac{17}{8}a=\frac{2}{-8}$$

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

$$a^{2}+\frac{17}{8}a=-\frac{1}{4}$$

Divide \frac{17}{8}, the coefficient of the x term, by 2 to get \frac{17}{16}. Then add the square of \frac{17}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{17}{8}a+\left(\frac{17}{16}\right)^{2}=-\frac{1}{4}+\left(\frac{17}{16}\right)^{2}$$

Square \frac{17}{16} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{17}{8}a+\frac{289}{256}=-\frac{1}{4}+\frac{289}{256}$$

Add -\frac{1}{4} to \frac{289}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+\frac{17}{8}a+\frac{289}{256}=\frac{225}{256}$$

Factor a^{2}+\frac{17}{8}a+\frac{289}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+\frac{17}{16}\right)^{2}=\frac{225}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{17}{16}\right)^{2}}=\sqrt{\frac{225}{256}}$$

Simplify.

$$a+\frac{17}{16}=\frac{15}{16} a+\frac{17}{16}=-\frac{15}{16}$$

Subtract \frac{17}{16} from both sides of the equation.

$$a=-\frac{1}{8} a=-2$$