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x^{2}-30x-24=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{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\left(-24\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -30 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-30\right)±\sqrt{900-4\left(-24\right)}}{2}
Square -30.
x=\frac{-\left(-30\right)±\sqrt{900+96}}{2}
Multiply -4 times -24.
x=\frac{-\left(-30\right)±\sqrt{996}}{2}
Add 900 to 96.
x=\frac{-\left(-30\right)±2\sqrt{249}}{2}
Take the square root of 996.
x=\frac{30±2\sqrt{249}}{2}
The opposite of -30 is 30.
x=\frac{2\sqrt{249}+30}{2}
Now solve the equation x=\frac{30±2\sqrt{249}}{2} when ± is plus. Add 30 to 2\sqrt{249}.
x=\sqrt{249}+15
Divide 30+2\sqrt{249} by 2.
x=\frac{30-2\sqrt{249}}{2}
Now solve the equation x=\frac{30±2\sqrt{249}}{2} when ± is minus. Subtract 2\sqrt{249} from 30.
x=15-\sqrt{249}
Divide 30-2\sqrt{249} by 2.
x=\sqrt{249}+15 x=15-\sqrt{249}
The equation is now solved.
x^{2}-30x-24=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.
x^{2}-30x-24-\left(-24\right)=-\left(-24\right)
Add 24 to both sides of the equation.
x^{2}-30x=-\left(-24\right)
Subtracting -24 from itself leaves 0.
x^{2}-30x=24
Subtract -24 from 0.
x^{2}-30x+\left(-15\right)^{2}=24+\left(-15\right)^{2}
Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-30x+225=24+225
Square -15.
x^{2}-30x+225=249
Add 24 to 225.
\left(x-15\right)^{2}=249
Factor x^{2}-30x+225. 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-15\right)^{2}}=\sqrt{249}
Take the square root of both sides of the equation.
x-15=\sqrt{249} x-15=-\sqrt{249}
Simplify.
x=\sqrt{249}+15 x=15-\sqrt{249}
Add 15 to both sides of the equation.
x ^ 2 -30x -24 = 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 = 30 rs = -24
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 = 15 - u s = 15 + u
Two numbers r and s sum up to 30 exactly when the average of the two numbers is \frac{1}{2}*30 = 15. 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>
(15 - u) (15 + u) = -24
To solve for unknown quantity u, substitute these in the product equation rs = -24
225 - u^2 = -24
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -24-225 = -249
Simplify the expression by subtracting 225 on both sides
u^2 = 249 u = \pm\sqrt{249} = \pm \sqrt{249}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =15 - \sqrt{249} = -0.780 s = 15 + \sqrt{249} = 30.780
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.