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y^{2}-8y+32=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\times 32}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
y=\frac{-\left(-8\right)±\sqrt{64-4\times 32}}{2}
Square -8.
y=\frac{-\left(-8\right)±\sqrt{64-128}}{2}
Multiply -4 times 32.
y=\frac{-\left(-8\right)±\sqrt{-64}}{2}
Add 64 to -128.
y=\frac{-\left(-8\right)±8i}{2}
Take the square root of -64.
y=\frac{8±8i}{2}
The opposite of -8 is 8.
y=\frac{8+8i}{2}
Now solve the equation y=\frac{8±8i}{2} when ± is plus. Add 8 to 8i.
y=4+4i
Divide 8+8i by 2.
y=\frac{8-8i}{2}
Now solve the equation y=\frac{8±8i}{2} when ± is minus. Subtract 8i from 8.
y=4-4i
Divide 8-8i by 2.
y=4+4i y=4-4i
The equation is now solved.
y^{2}-8y+32=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+32-32=-32
Subtract 32 from both sides of the equation.
y^{2}-8y=-32
Subtracting 32 from itself leaves 0.
y^{2}-8y+\left(-4\right)^{2}=-32+\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=-32+16
Square -4.
y^{2}-8y+16=-16
Add -32 to 16.
\left(y-4\right)^{2}=-16
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{-16}
Take the square root of both sides of the equation.
y-4=4i y-4=-4i
Simplify.
y=4+4i y=4-4i
Add 4 to both sides of the equation.
x ^ 2 -8x +32 = 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 = 32
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) = 32
To solve for unknown quantity u, substitute these in the product equation rs = 32
16 - u^2 = 32
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 32-16 = 16
Simplify the expression by subtracting 16 on both sides
u^2 = -16 u = \pm\sqrt{-16} = \pm 4i
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =4 - 4i s = 4 + 4i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.