Skip to main content
Solve for x
Tick mark Image
Graph

Similar Problems from Web Search

Share

a+b=-8 ab=3\times 5=15
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3x^{2}+ax+bx+5. To find a and b, set up a system to be solved.
-1,-15 -3,-5
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 15.
-1-15=-16 -3-5=-8
Calculate the sum for each pair.
a=-5 b=-3
The solution is the pair that gives sum -8.
\left(3x^{2}-5x\right)+\left(-3x+5\right)
Rewrite 3x^{2}-8x+5 as \left(3x^{2}-5x\right)+\left(-3x+5\right).
x\left(3x-5\right)-\left(3x-5\right)
Factor out x in the first and -1 in the second group.
\left(3x-5\right)\left(x-1\right)
Factor out common term 3x-5 by using distributive property.
x=\frac{5}{3} x=1
To find equation solutions, solve 3x-5=0 and x-1=0.
3x^{2}-8x+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(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 3\times 5}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -8 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-8\right)±\sqrt{64-4\times 3\times 5}}{2\times 3}
Square -8.
x=\frac{-\left(-8\right)±\sqrt{64-12\times 5}}{2\times 3}
Multiply -4 times 3.
x=\frac{-\left(-8\right)±\sqrt{64-60}}{2\times 3}
Multiply -12 times 5.
x=\frac{-\left(-8\right)±\sqrt{4}}{2\times 3}
Add 64 to -60.
x=\frac{-\left(-8\right)±2}{2\times 3}
Take the square root of 4.
x=\frac{8±2}{2\times 3}
The opposite of -8 is 8.
x=\frac{8±2}{6}
Multiply 2 times 3.
x=\frac{10}{6}
Now solve the equation x=\frac{8±2}{6} when ± is plus. Add 8 to 2.
x=\frac{5}{3}
Reduce the fraction \frac{10}{6} to lowest terms by extracting and canceling out 2.
x=\frac{6}{6}
Now solve the equation x=\frac{8±2}{6} when ± is minus. Subtract 2 from 8.
x=1
Divide 6 by 6.
x=\frac{5}{3} x=1
The equation is now solved.
3x^{2}-8x+5=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.
3x^{2}-8x+5-5=-5
Subtract 5 from both sides of the equation.
3x^{2}-8x=-5
Subtracting 5 from itself leaves 0.
\frac{3x^{2}-8x}{3}=-\frac{5}{3}
Divide both sides by 3.
x^{2}-\frac{8}{3}x=-\frac{5}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}-\frac{8}{3}x+\left(-\frac{4}{3}\right)^{2}=-\frac{5}{3}+\left(-\frac{4}{3}\right)^{2}
Divide -\frac{8}{3}, the coefficient of the x term, by 2 to get -\frac{4}{3}. Then add the square of -\frac{4}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{8}{3}x+\frac{16}{9}=-\frac{5}{3}+\frac{16}{9}
Square -\frac{4}{3} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{8}{3}x+\frac{16}{9}=\frac{1}{9}
Add -\frac{5}{3} to \frac{16}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{4}{3}\right)^{2}=\frac{1}{9}
Factor x^{2}-\frac{8}{3}x+\frac{16}{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}.
\sqrt{\left(x-\frac{4}{3}\right)^{2}}=\sqrt{\frac{1}{9}}
Take the square root of both sides of the equation.
x-\frac{4}{3}=\frac{1}{3} x-\frac{4}{3}=-\frac{1}{3}
Simplify.
x=\frac{5}{3} x=1
Add \frac{4}{3} to both sides of the equation.