Solve for y
y=-1
y=9
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a+b=-8 ab=-9
To solve the equation, factor y^{2}-8y-9 using formula y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right). To find a and b, set up a system to be solved.
1,-9 3,-3
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 -9.
1-9=-8 3-3=0
Calculate the sum for each pair.
a=-9 b=1
The solution is the pair that gives sum -8.
\left(y-9\right)\left(y+1\right)
Rewrite factored expression \left(y+a\right)\left(y+b\right) using the obtained values.
y=9 y=-1
To find equation solutions, solve y-9=0 and y+1=0.
a+b=-8 ab=1\left(-9\right)=-9
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as y^{2}+ay+by-9. To find a and b, set up a system to be solved.
1,-9 3,-3
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 -9.
1-9=-8 3-3=0
Calculate the sum for each pair.
a=-9 b=1
The solution is the pair that gives sum -8.
\left(y^{2}-9y\right)+\left(y-9\right)
Rewrite y^{2}-8y-9 as \left(y^{2}-9y\right)+\left(y-9\right).
y\left(y-9\right)+y-9
Factor out y in y^{2}-9y.
\left(y-9\right)\left(y+1\right)
Factor out common term y-9 by using distributive property.
y=9 y=-1
To find equation solutions, solve y-9=0 and y+1=0.
y^{2}-8y-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.
y=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-9\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
y=\frac{-\left(-8\right)±\sqrt{64-4\left(-9\right)}}{2}
Square -8.
y=\frac{-\left(-8\right)±\sqrt{64+36}}{2}
Multiply -4 times -9.
y=\frac{-\left(-8\right)±\sqrt{100}}{2}
Add 64 to 36.
y=\frac{-\left(-8\right)±10}{2}
Take the square root of 100.
y=\frac{8±10}{2}
The opposite of -8 is 8.
y=\frac{18}{2}
Now solve the equation y=\frac{8±10}{2} when ± is plus. Add 8 to 10.
y=9
Divide 18 by 2.
y=-\frac{2}{2}
Now solve the equation y=\frac{8±10}{2} when ± is minus. Subtract 10 from 8.
y=-1
Divide -2 by 2.
y=9 y=-1
The equation is now solved.
y^{2}-8y-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.
y^{2}-8y-9-\left(-9\right)=-\left(-9\right)
Add 9 to both sides of the equation.
y^{2}-8y=-\left(-9\right)
Subtracting -9 from itself leaves 0.
y^{2}-8y=9
Subtract -9 from 0.
y^{2}-8y+\left(-4\right)^{2}=9+\left(-4\right)^{2}
Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
y^{2}-8y+16=9+16
Square -4.
y^{2}-8y+16=25
Add 9 to 16.
\left(y-4\right)^{2}=25
Factor y^{2}-8y+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(y-4\right)^{2}}=\sqrt{25}
Take the square root of both sides of the equation.
y-4=5 y-4=-5
Simplify.
y=9 y=-1
Add 4 to both sides of the equation.
x ^ 2 -8x -9 = 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.
r + s = 8 rs = -9
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 = 4 - u s = 4 + u
Two numbers r and s sum up to 8 exactly when the average of the two numbers is \frac{1}{2}*8 = 4. 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>
(4 - u) (4 + u) = -9
To solve for unknown quantity u, substitute these in the product equation rs = -9
16 - u^2 = -9
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -9-16 = -25
Simplify the expression by subtracting 16 on both sides
u^2 = 25 u = \pm\sqrt{25} = \pm 5
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =4 - 5 = -1 s = 4 + 5 = 9
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
Examples
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{ 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}