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Quadratic equation solver
This calculator solves quadratic equations using three different methods : the quadratic formula method, completing the square, and the factoring method. Calculator shows all the work and provides detailed explanation on how to solve an equation.
Solve $\color{blue}{x^2-3 = 0}$ using the Quadratic Formula.
solution
$$ \color{blue}{ x_1 = -\sqrt{ 3 } }~~ \text{and}~~ \color{blue}{ x_2 = \sqrt{ 3 } } $$explanation
Step 1: Read the values of $ a $, $ b $, and $ c $ from the quadratic equation: $ a $ is the number in front of $ x^2 $, $ b $ is the number in front of $ x $, $ c $ is the number at the end. In our case:
$$ a = 1, \,\, b = 0, \,\, c = -3 $$Step 2: Plug in the values for $ a $, $ b $, and $ c $ into the quadratic formula.
$$ \begin{aligned} x_1,x_2 &= \frac{-b \pm \sqrt{b^2-4ac}}{2a} \\[1 em] x_1,x_2 &= \frac{ -0 \pm \sqrt{ 0 ^2 - 4 \cdot 1 \cdot (-3)} }{ 2 \cdot 1 } \end{aligned} $$Step 3: Simplify expression under the square root.
$$ x_1,x_2 = \frac{ 0 \pm \sqrt{ 12 } }{ 2 } $$Step 4: Solve for $ x $
$$ \begin{aligned} & \color{blue}{ x_1 = \frac{ 0~-~\sqrt{ 12 } }{ 2 } = -\sqrt{ 3 } } \\\\ & \color{blue}{ x_2 = \frac{ 0~+~\sqrt{ 12 } }{ 2 } = \sqrt{ 3 } } \end{aligned} $$THE ALTERNATIVE SOLUTION:
Isolate the squared variable term:
$$ x^2 = 3 $$Solve for x:
$$ x_1 = \sqrt { 3 } $$$$ x_2 = -\sqrt { 3 } $$How to use this calculator
The most commonly used methods for solving quadratic equations are:
1. Factoring method
2. Solving quadratic equations by completing the square
3. Using quadratic formula
In the following sections, we'll go over these methods.
Method 1A : Factoring method
If a quadratic trinomial can be factored, this is the best solving method.
We often use this method when the leading coefficient is equal to 1 or -1. If this is not the case, then it is better to use some other method.
Example 01: Solve x2-8x+15=0 by factoring.
Here we see that the leading coefficient is 1, so the factoring method is our first choice.
To factor this equation, we must find two numbers a and b with a sum of a + b = 8 and a product of a × b = 15
After some trials and errors, we see that a = 3 and b = 5.
Now we use a formula x2-8x+15=(x-a)(x-b) to get factored form:
x2-8x+15=(x-3)(x-5)
Divide the factored form into two linear equations to get solutions.
$$ \begin{aligned} x^2 - 8x + 15 &= 0 \\ (x - 3)(x - 5) &= 0 \\ x -3 &= 0 ~~ \text{or} ~~ x - 5= 0 \\ x &= 3 ~~ \text{or} ~~ x = 5 \end{aligned} $$Method 1B: Factoring – special cases
Example 02: Solve x2-8x=0 by factoring.
In this case, (when the coefficient c = 0) we can factor out x out of x2-8.
$$ \begin{aligned} x^2 - 8x &= 0 \\ \color{blue}{x} \cdot ( x - 8 ) &= 0 \\ x &= 0 ~~ \text{or} ~~ x - 8 = 0 \\ x &= 0 ~~ \text{or} ~~ x = 8 \end{aligned} $$Example 03: Solve x2-16=0 by factoring.
In this case, (when the middle term is equal 0) we can use the difference of squares formula.
$$ \begin{aligned} x^2 - 16 &= 0 \\ x^2 - 4^2 &= 0 \text{ use } a^2 - b^2 = (a-b)(a+b) \\ (x - 4)(x+4) &= 0 \\ x - 4 &= 0 ~~ \text{or} ~~ x + 4 = 0 \\ x &= 4 ~~ \text{or} ~~ x = -4 \end{aligned} $$Method 3 : Solve using quadratic formula
This method solves all types of quadratic equations. It works best when solutions contain some radicals or complex numbers.
Example 05: Solve equation $ 2x^2 + 3x - 2 = 0$ by using quadratic formula.
Step 1: Read the values of $a$, $b$, and $c$ from the quadratic equation. (a is the number in front of x2, b is the number in front of x and c is the number at the end)
a = 2, b = 3 and c = -2
Step 2:Plug the values for a, b, and c into the quadratic formula and simplify.
$$ \begin{aligned} x_1, x_2 &= \frac{-b \pm \sqrt{b^2-4ac}}{2a} \\ x_1, x_2 &= \frac{-3 \pm \sqrt{3^2-4 \cdot 2 \cdot (-2) }}{2 \cdot 2} \\ x_1, x_2 &= \frac{-3 \pm \sqrt{9+ 16 }}{4} \\ x_1, x_2 &= \frac{-3 \pm \sqrt{25}}{4} \\ x_1, x_2 &= \frac{-3 \pm 5}{4} \end{aligned} $$Step 3: Solve for x1 and x2
$$ \begin{aligned} x_1 = & \frac{-3 \color{blue}{+} 5}{4} = \frac{2}{4} = \frac{1}{2} \\ x_2 = & \frac{-3 \color{blue}{-} 5}{4} = \frac{-8}{4} = -2 \end{aligned} $$Method 2 : Completing the square
This method can be used to solve all types of quadratic equations, although it can be complicated for some types of equations. The method involves seven steps.
Example 04: Solve equation 2x2+8x-10=0 by completing the square.
Step 1: Divide the equation by the number in front of the square term.
$$ \begin{aligned} 2x^2 + 8x - 10 & = 0 ~~ / ~ \color{orangered}{:2} \\ \frac{2x^2}{2} + \frac{8x}{2} - \frac{10}{2} & = \frac{0}{2} \\ x^2 + 4x - 5 & = 0 \end{aligned} $$Step 2: move -5 to the right:
x2+4x=5
Step 3: Take half of the x-term coefficient $ \color{blue}{\dfrac{4}{2}} $, square it $ \color{blue}{\left(\dfrac{4}{2} \right)^2} $ and add this value to both sides.
$$ x^2 + 4x + \color{blue}{\left(\frac{4}{2} \right)^2} = 5 + \color{blue}{\left(\frac{4}{2} \right)^2} $$Step 4: Simplify left and right side.
x2+4x+22=9
Step 5: Write the perfect square on the left.
$$ \left( x + 2 \right)^2 = 9 $$Step 6: Take the square root of both sides.
$$ \begin{aligned} x + 2 &= \pm \sqrt{9} \\\\ x + 2 &= \pm 3 \end{aligned} $$Step 7: Solve for $x_1$ and $x_2$ .
$$ \begin{aligned} x_1 & = +3 - 2 = 1 \\ x_2 & = -3 - 2 = - 5 \end{aligned} $$1. Quadratic Equation — step-by-step examples, video tutorials with worked examples.
2. Completing the Square — video on Khan Academy
3. Completing the Square — video on Khan Academy