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a+b=10 ab=16\left(-9\right)=-144
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 16x^{2}+ax+bx-9. To find a and b, set up a system to be solved.
-1,144 -2,72 -3,48 -4,36 -6,24 -8,18 -9,16 -12,12
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -144.
-1+144=143 -2+72=70 -3+48=45 -4+36=32 -6+24=18 -8+18=10 -9+16=7 -12+12=0
Calculate the sum for each pair.
a=-8 b=18
The solution is the pair that gives sum 10.
\left(16x^{2}-8x\right)+\left(18x-9\right)
Rewrite 16x^{2}+10x-9 as \left(16x^{2}-8x\right)+\left(18x-9\right).
8x\left(2x-1\right)+9\left(2x-1\right)
Factor out 8x in the first and 9 in the second group.
\left(2x-1\right)\left(8x+9\right)
Factor out common term 2x-1 by using distributive property.
x=\frac{1}{2} x=-\frac{9}{8}
To find equation solutions, solve 2x-1=0 and 8x+9=0.
16x^{2}+10x-9=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{-10±\sqrt{10^{2}-4\times 16\left(-9\right)}}{2\times 16}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 10 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-10±\sqrt{100-4\times 16\left(-9\right)}}{2\times 16}
Square 10.
x=\frac{-10±\sqrt{100-64\left(-9\right)}}{2\times 16}
Multiply -4 times 16.
x=\frac{-10±\sqrt{100+576}}{2\times 16}
Multiply -64 times -9.
x=\frac{-10±\sqrt{676}}{2\times 16}
Add 100 to 576.
x=\frac{-10±26}{2\times 16}
Take the square root of 676.
x=\frac{-10±26}{32}
Multiply 2 times 16.
x=\frac{16}{32}
Now solve the equation x=\frac{-10±26}{32} when ± is plus. Add -10 to 26.
x=\frac{1}{2}
Reduce the fraction \frac{16}{32} to lowest terms by extracting and canceling out 16.
x=-\frac{36}{32}
Now solve the equation x=\frac{-10±26}{32} when ± is minus. Subtract 26 from -10.
x=-\frac{9}{8}
Reduce the fraction \frac{-36}{32} to lowest terms by extracting and canceling out 4.
x=\frac{1}{2} x=-\frac{9}{8}
The equation is now solved.
16x^{2}+10x-9=0
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.
16x^{2}+10x-9-\left(-9\right)=-\left(-9\right)
Add 9 to both sides of the equation.
16x^{2}+10x=-\left(-9\right)
Subtracting -9 from itself leaves 0.
16x^{2}+10x=9
Subtract -9 from 0.
\frac{16x^{2}+10x}{16}=\frac{9}{16}
Divide both sides by 16.
x^{2}+\frac{10}{16}x=\frac{9}{16}
Dividing by 16 undoes the multiplication by 16.
x^{2}+\frac{5}{8}x=\frac{9}{16}
Reduce the fraction \frac{10}{16} to lowest terms by extracting and canceling out 2.
x^{2}+\frac{5}{8}x+\left(\frac{5}{16}\right)^{2}=\frac{9}{16}+\left(\frac{5}{16}\right)^{2}
Divide \frac{5}{8}, the coefficient of the x term, by 2 to get \frac{5}{16}. Then add the square of \frac{5}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{5}{8}x+\frac{25}{256}=\frac{9}{16}+\frac{25}{256}
Square \frac{5}{16} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{5}{8}x+\frac{25}{256}=\frac{169}{256}
Add \frac{9}{16} to \frac{25}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{5}{16}\right)^{2}=\frac{169}{256}
Factor x^{2}+\frac{5}{8}x+\frac{25}{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}.
\sqrt{\left(x+\frac{5}{16}\right)^{2}}=\sqrt{\frac{169}{256}}
Take the square root of both sides of the equation.
x+\frac{5}{16}=\frac{13}{16} x+\frac{5}{16}=-\frac{13}{16}
Simplify.
x=\frac{1}{2} x=-\frac{9}{8}
Subtract \frac{5}{16} from both sides of the equation.
x ^ 2 +\frac{5}{8}x -\frac{9}{16} = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 16
r + s = -\frac{5}{8} rs = -\frac{9}{16}
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = -\frac{5}{16} - u s = -\frac{5}{16} + u
Two numbers r and s sum up to -\frac{5}{8} exactly when the average of the two numbers is \frac{1}{2}*-\frac{5}{8} = -\frac{5}{16}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(-\frac{5}{16} - u) (-\frac{5}{16} + u) = -\frac{9}{16}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{9}{16}
\frac{25}{256} - u^2 = -\frac{9}{16}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{9}{16}-\frac{25}{256} = -\frac{169}{256}
Simplify the expression by subtracting \frac{25}{256} on both sides
u^2 = \frac{169}{256} u = \pm\sqrt{\frac{169}{256}} = \pm \frac{13}{16}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{5}{16} - \frac{13}{16} = -1.125 s = -\frac{5}{16} + \frac{13}{16} = 0.500
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.