Solve for b
b = -\frac{5}{4} = -1\frac{1}{4} = -1.25
b=-\frac{1}{4}=-0.25
Share
Copied to clipboard
16b^{2}+24b+20-15=0
Subtract 15 from both sides.
16b^{2}+24b+5=0
Subtract 15 from 20 to get 5.
a+b=24 ab=16\times 5=80
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 16b^{2}+ab+bb+5. To find a and b, set up a system to be solved.
1,80 2,40 4,20 5,16 8,10
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 80.
1+80=81 2+40=42 4+20=24 5+16=21 8+10=18
Calculate the sum for each pair.
a=4 b=20
The solution is the pair that gives sum 24.
\left(16b^{2}+4b\right)+\left(20b+5\right)
Rewrite 16b^{2}+24b+5 as \left(16b^{2}+4b\right)+\left(20b+5\right).
4b\left(4b+1\right)+5\left(4b+1\right)
Factor out 4b in the first and 5 in the second group.
\left(4b+1\right)\left(4b+5\right)
Factor out common term 4b+1 by using distributive property.
b=-\frac{1}{4} b=-\frac{5}{4}
To find equation solutions, solve 4b+1=0 and 4b+5=0.
16b^{2}+24b+20=15
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.
16b^{2}+24b+20-15=15-15
Subtract 15 from both sides of the equation.
16b^{2}+24b+20-15=0
Subtracting 15 from itself leaves 0.
16b^{2}+24b+5=0
Subtract 15 from 20.
b=\frac{-24±\sqrt{24^{2}-4\times 16\times 5}}{2\times 16}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 24 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
b=\frac{-24±\sqrt{576-4\times 16\times 5}}{2\times 16}
Square 24.
b=\frac{-24±\sqrt{576-64\times 5}}{2\times 16}
Multiply -4 times 16.
b=\frac{-24±\sqrt{576-320}}{2\times 16}
Multiply -64 times 5.
b=\frac{-24±\sqrt{256}}{2\times 16}
Add 576 to -320.
b=\frac{-24±16}{2\times 16}
Take the square root of 256.
b=\frac{-24±16}{32}
Multiply 2 times 16.
b=-\frac{8}{32}
Now solve the equation b=\frac{-24±16}{32} when ± is plus. Add -24 to 16.
b=-\frac{1}{4}
Reduce the fraction \frac{-8}{32} to lowest terms by extracting and canceling out 8.
b=-\frac{40}{32}
Now solve the equation b=\frac{-24±16}{32} when ± is minus. Subtract 16 from -24.
b=-\frac{5}{4}
Reduce the fraction \frac{-40}{32} to lowest terms by extracting and canceling out 8.
b=-\frac{1}{4} b=-\frac{5}{4}
The equation is now solved.
16b^{2}+24b+20=15
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.
16b^{2}+24b+20-20=15-20
Subtract 20 from both sides of the equation.
16b^{2}+24b=15-20
Subtracting 20 from itself leaves 0.
16b^{2}+24b=-5
Subtract 20 from 15.
\frac{16b^{2}+24b}{16}=-\frac{5}{16}
Divide both sides by 16.
b^{2}+\frac{24}{16}b=-\frac{5}{16}
Dividing by 16 undoes the multiplication by 16.
b^{2}+\frac{3}{2}b=-\frac{5}{16}
Reduce the fraction \frac{24}{16} to lowest terms by extracting and canceling out 8.
b^{2}+\frac{3}{2}b+\left(\frac{3}{4}\right)^{2}=-\frac{5}{16}+\left(\frac{3}{4}\right)^{2}
Divide \frac{3}{2}, the coefficient of the x term, by 2 to get \frac{3}{4}. Then add the square of \frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
b^{2}+\frac{3}{2}b+\frac{9}{16}=\frac{-5+9}{16}
Square \frac{3}{4} by squaring both the numerator and the denominator of the fraction.
b^{2}+\frac{3}{2}b+\frac{9}{16}=\frac{1}{4}
Add -\frac{5}{16} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(b+\frac{3}{4}\right)^{2}=\frac{1}{4}
Factor b^{2}+\frac{3}{2}b+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(b+\frac{3}{4}\right)^{2}}=\sqrt{\frac{1}{4}}
Take the square root of both sides of the equation.
b+\frac{3}{4}=\frac{1}{2} b+\frac{3}{4}=-\frac{1}{2}
Simplify.
b=-\frac{1}{4} b=-\frac{5}{4}
Subtract \frac{3}{4} from both sides of the equation.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}