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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 $\color{blue}{x^2+2x-8 = 0}$ using factoring.

**solution**

**explanation**

First we need to factor trinomial $ \color{blue}{ x^2+2x-8 } $ and than we use factored form to solve an equation $ \color{blue}{ x^2+2x-8 = 0} $.

** Step 1:** Identify constants $ \color{blue}{ b }$ and $\color{red}{ c }$.
( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:

Now we must discover two numbers that sum up to $ \color{blue}{ 2 } $ and multiply to $ \color{red}{ -8 } $.

** Step 2:** Find out pairs of numbers with a product of $\color{red}{ c = -8 }$.

PRODUCT = -8 | |

-1 8 | 1 -8 |

-2 4 | 2 -4 |

** Step 3:** Find out which pair sums up to $\color{blue}{ b = 2 }$

PRODUCT = -8 and SUM = 2 | |

-1 8 | 1 -8 |

-2 4 | 2 -4 |

** Step 4:** Put -2 and 4 into placeholders to get factored form.

** Step 5:** Set each factor to zero and solve equations.

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Examples

ex 1:

Solve x^{2}+3x-4=0 by factoring.

ex 2:

Solve 4x^{2}-x-3=0 by completing the square.

ex 3:

Solve -2x^{2}-0.5x+0.75=0 using the quadratic
formula.

ex 4:

Solve 2/3x^{2}-1/3x-5=0.

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Find more worked-out examples in our database of solved problems..

TUTORIAL

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.

**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 x^{2}-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 x^{2}-8x+15=(x-a)(x-b)
to get factored form:

x^{2}-8x+15=(x-3)(x-5)

Divide the factored form into two linear equations to get solutions.

$$ \begin{aligned} x^2 - 8x + 15 &= 0 \\ (x - 3)(x - 5) &= 0 \\ x -3 &= 0 ~~ \text{or} ~~ x - 5= 0 \\ x &= 3 ~~ \text{or} ~~ x = 5 \end{aligned} $$**Example 02:** Solve x^{2}-8x=0 by factoring.

In this case, (when the coefficient c = 0) we can factor out
x out of x^{2}-8.

**Example 03:** Solve x^{2}-16=0 by factoring.

In this case, (when the middle term is equal 0) we can use the difference of squares formula.

$$ \begin{aligned} x^2 - 16 &= 0 \\ x^2 - 4^2 &= 0 \text{ use } a^2 - b^2 = (a-b)(a+b) \\ (x - 4)(x+4) &= 0 \\ x - 4 &= 0 ~~ \text{or} ~~ x + 4 = 0 \\ x &= 4 ~~ \text{or} ~~ x = -4 \end{aligned} $$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 x^{2},
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.

**Step 3**: Solve for x_{1} and x_{2}

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 2x^{2}+8x-10=0 by completing the
square.

**Step 1**: Divide the equation by the number in front of the square
term.

**Step 2**: move -5 to the right:

x^{2}+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.

**Step 4**: Simplify left and right side.

x^{2}+4x+2^{2}=9

**Step 5**: Write the perfect square on the left.

**Step 6**: Take the square root of both sides.

**Step 7**: Solve for $x_1$ and $x_2$ .

RESOURCES

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

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