Algebra Examples

$2 x (x - 3) = x - 5$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $2 x (x - 3)$ .

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Step 1.1.1.1

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 3 = x - 5$

Step 1.1.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.1.2.1

Move $x$ .

$2 (x \cdot x) + 2 x \cdot - 3 = x - 5$

Step 1.1.1.2.2

Multiply $x$ by $x$ .

$2 x^{2} + 2 x \cdot - 3 = x - 5$

Step 1.1.1.3

Multiply $- 3$ by $2$ .

$2 x^{2} - 6 x = x - 5$

Step 1.2

Move all the expressions to the left side of the equation.

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Step 1.2.1

Subtract $x$ from both sides of the equation.

$2 x^{2} - 6 x - x = - 5$

Step 1.2.2

Add $5$ to both sides of the equation.

$2 x^{2} - 6 x - x + 5 = 0$

Step 1.3

Subtract $x$ from $- 6 x$ .

$2 x^{2} - 7 x + 5 = 0$

Step 2

The discriminant of a quadratic is the expression inside the radical of the quadratic formula.

$b^{2} - 4 (a c)$

Step 3

Substitute in the values of $a$ , $b$ , and $c$ .

${(- 7)}^{2} - 4 (2 \cdot 5)$

Step 4

Evaluate the result to find the discriminant.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raise $- 7$ to the power of $2$ .

$49 - 4 (2 \cdot 5)$

Step 4.1.2

Multiply $- 4 (2 \cdot 5)$ .

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Step 4.1.2.1

Multiply $2$ by $5$ .

$49 - 4 \cdot 10$

Step 4.1.2.2

Multiply $- 4$ by $10$ .

$49 - 40$

Step 4.2

Subtract $40$ from $49$ .

$9$

Step 5

The nature of the roots of the quadratic can fall into one of three categories depending on the value of the discriminant $(Δ)$ :

$Δ > 0$ means there are $2$ distinct real roots.

$Δ = 0$ means there are $2$ equal real roots, or $1$ distinct real root.

$Δ < 0$ means there are no real roots, but $2$ complex roots.

Since the discriminant is greater than $0$ , there are two real roots.

Two Real Roots