Solve for x
Steps Using Factoring By Grouping
Steps Using the Quadratic Formula
Steps for Completing the Square
Steps Using Direct Factoring Method
Graph Both Sides in 2D
Graph in 2D
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a+b=-30 ab=9\times 25=225
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 9x^{2}+ax+bx+25. To find a and b, set up a system to be solved.
-1,-225 -3,-75 -5,-45 -9,-25 -15,-15
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 225.
-1-225=-226 -3-75=-78 -5-45=-50 -9-25=-34 -15-15=-30
Calculate the sum for each pair.
a=-15 b=-15
The solution is the pair that gives sum -30.
Rewrite 9x^{2}-30x+25 as \left(9x^{2}-15x\right)+\left(-15x+25\right).
Factor out 3x in the first and -5 in the second group.
Factor out common term 3x-5 by using distributive property.
Rewrite as a binomial square.
To find equation solution, solve 3x-5=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\times 9\times 25}}{2\times 9}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -30 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-30\right)±\sqrt{900-4\times 9\times 25}}{2\times 9}
Square -30.
x=\frac{-\left(-30\right)±\sqrt{900-36\times 25}}{2\times 9}
Multiply -4 times 9.
x=\frac{-\left(-30\right)±\sqrt{900-900}}{2\times 9}
Multiply -36 times 25.
x=\frac{-\left(-30\right)±\sqrt{0}}{2\times 9}
Add 900 to -900.
x=-\frac{-30}{2\times 9}
Take the square root of 0.
x=\frac{30}{2\times 9}
The opposite of -30 is 30.
Multiply 2 times 9.
Reduce the fraction \frac{30}{18}\approx 1.666666667 to lowest terms by extracting and canceling out 6.
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.
Subtract 25 from both sides of the equation.
Subtracting 25 from itself leaves 0.
Divide both sides by 9.
Dividing by 9 undoes the multiplication by 9.
Reduce the fraction \frac{-30}{9}\approx -3.333333333 to lowest terms by extracting and canceling out 3.
Divide -25 by 9.
Divide -\frac{10}{3}\approx -3.333333333, the coefficient of the x term, by 2 to get -\frac{5}{3}\approx -1.666666667. Then add the square of -\frac{5}{3}\approx -1.666666667 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
Square -\frac{5}{3}\approx -1.666666667 by squaring both the numerator and the denominator of the fraction.
Add -\frac{25}{9}\approx -2.777777778 to \frac{25}{9}\approx 2.777777778 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
Factor x^{2}-\frac{10}{3}x+\frac{25}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
Take the square root of both sides of the equation.
x-\frac{5}{3}=0 x-\frac{5}{3}=0
x=\frac{5}{3} x=\frac{5}{3}
Add \frac{5}{3}\approx 1.666666667 to both sides of the equation.
The equation is now solved. Solutions are the same.
x ^ 2 -\frac{10}{3}x +\frac{25}{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.This is achieved by dividing both sides of the equation by 9
r + s = \frac{10}{3} rs = \frac{25}{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 = \frac{5}{3} - u s = \frac{5}{3} + u
Two numbers r and s sum up to \frac{10}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{10}{3} = \frac{5}{3}. 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='' style='width: 100%;max-width: 700px' /></div>
(\frac{5}{3} - u) (\frac{5}{3} + u) = \frac{25}{9}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{25}{9}
\frac{25}{9} - u^2 = \frac{25}{9}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{25}{9}-\frac{25}{9} = 0
Simplify the expression by subtracting \frac{25}{9} on both sides
u^2 = 0 u = 0
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r = s = \frac{5}{3} = 1.667
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.