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Solution - Solving quadratic equations by factoring

Exact form:

x_{1} = - \frac{1}{2}, x_{2} = 3

x_1=-\frac{1}{2}, x_2=3

Decimal form:

x_{1} = - 0.5, x_{2} = 3

x_1=-0.5, x_2=3

Equation in factored form:

(2 x + 1) (x - 3) = 0

(2x+1)(x-3)=0

See steps

Step-by-step explanation

1. Find the coefficients

To find the coefficients, use the standard form of a quadratic equation:
$a x^{2} + b x + c = 0$
$2 x^{2} - 5 x - 3 = 0$

Coefficient $a$ $= 2$
Coefficient $b$ $= - 5$
Coefficient $c$ $= - 3$

2. Find two numbers whose product equals $a \cdot c$ and sum equals $b$

Find the factors whose product equals coefficient $a$ multiplied by coefficient $c$ :
coefficient $a$ ∙ coefficient $c$ = $2$ ∙ $- 3$ = $- 6$

List the factors of $- 6$ :
$1, 2, 3, 6$

Because the product of coefficient $a$ and coefficient $c$ equals a negative number $(- 6)$ one factor needs to be positive and the other one negative.

From the list of factors, find a pair whose sum equals coefficient $b$ .
Coefficient $b$ = $- 5$

Found it - this pair does the trick:

$1 \cdot - 6 = - 6$

$1 - 6 = - 5$

The product of $1$ and $- 6$ equals coefficient $a$ $(2)$ multiplied by coefficient $c$ $(- 3)$ and their sum equals coefficient $b$ $(- 5)$ .

3. Split the middle term of the equation

$2 x^{2} - 5 x - 3 = 0$
Rewrite the middle term using $1$ and $- 6$ :
$2 x^{2} + x - 6 x - 3 = 0$

4. Factor by grouping

$2 x^{2} + x - 6 x - 3 = 0$

Factor out the first two terms and last two terms separately:

$(2 x^{2} + x) + (- 6 x - 3) = 0$

Factor out the first term:

$x (2 x + 1) + (- 6 x - 3) = 0$

Factor out the second term:

$x (2 x + 1) - 3 (2 x + 1) = 0$

Factor out the greatest common factor from each group:

$(2 x + 1) (x - 3) = 0$

The factors of $2 x^{2} - 5 x - 3 = 0$ are $(2 x + 1)$ and $(x - 3)$ .

5. Find the roots of the quadratic equation

If
$(2 x + 1)$ ∙ $(x - 3) = 0$
Then
$(2 x + 1) = 0$
and/or
$(x - 3) = 0$

Solve each factor for $x$ :

Factor 1:

4 additional steps

$(2 x + 1) = 0$

Subtract from both sides:

$(2 x + 1) - 1 = 0 - 1$

Simplify the arithmetic:

$2 x = 0 - 1$

Simplify the arithmetic:

$2 x = - 1$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{- 1}{2}$

Simplify the fraction:

$x = \frac{- 1}{2}$

Factor 2:

2 additional steps

$(x - 3) = 0$

Add to both sides:

$(x - 3) + 3 = 0 + 3$

Simplify the arithmetic:

$x = 0 + 3$

Simplify the arithmetic:

$x = 3$

6. Graph

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Why learn this

In their most basic function, quadratic equations define shapes like circles, ellipses and parabolas. These shapes can, in turn, be used to predict the curve of an object in motion, such as a ball kicked by a football player or a shot fired out of a cannon.
When it comes to an object’s movement through space, what better place to start than space itself, with the revolution of planets around the sun in our solar system? The quadratic equation was used to establish that planets’ orbits are elliptical, not circular. Determining the path and speed an object travels through space is possible even after it has come to a stop: the quadratic equation can calculate how fast a vehicle was moving when it crashed. With information like this, the automotive industry can design brakes to prevent collisions in the future. Many industries use the quadratic equation to predict and thus improve their products’ lifespan and safety.

Terms and topics

Solving quadratic equations by factoring

Solution - Solving quadratic equations by factoring

Step-by-step explanation

1. Find the coefficients

2. Find two numbers whose product equals a·c and sum equals b

3. Split the middle term of the equation

4. Factor by grouping

5. Find the roots of the quadratic equation

6. Graph

Why learn this

Terms and topics

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2. Find two numbers whose product equals $a \cdot c$ and sum equals $b$