Finite Math Examples

$x (x + 3) - 2 = 3 x + 23$

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 each term.

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Step 1.1.1.1

Apply the distributive property.

$x \cdot x + x \cdot 3 - 2 = 3 x + 23$

Step 1.1.1.2

Multiply $x$ by $x$ .

$x^{2} + x \cdot 3 - 2 = 3 x + 23$

Step 1.1.1.3

Move $3$ to the left of $x$ .

$x^{2} + 3 x - 2 = 3 x + 23$

Step 1.2

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

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Step 1.2.1

Subtract $3 x$ from both sides of the equation.

$x^{2} + 3 x - 2 - 3 x = 23$

Step 1.2.2

Subtract $23$ from both sides of the equation.

$x^{2} + 3 x - 2 - 3 x - 23 = 0$

Step 1.3

Simplify $x^{2} + 3 x - 2 - 3 x - 23$ .

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Step 1.3.1

Combine the opposite terms in $x^{2} + 3 x - 2 - 3 x - 23$ .

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Step 1.3.1.1

Subtract $3 x$ from $3 x$ .

$x^{2} + 0 - 2 - 23 = 0$

Step 1.3.1.2

Add $x^{2}$ and $0$ .

$x^{2} - 2 - 23 = 0$

Step 1.3.2

Subtract $23$ from $- 2$ .

$x^{2} - 25 = 0$

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 $c$ .

$0^{2} - 4 (1 \cdot - 25)$

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

Raising $0$ to any positive power yields $0$ .

$0 - 4 (1 \cdot - 25)$

Step 4.1.2

Multiply $- 4 (1 \cdot - 25)$ .

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Step 4.1.2.1

Multiply $- 25$ by $1$ .

$0 - 4 \cdot - 25$

Step 4.1.2.2

Multiply $- 4$ by $- 25$ .

$0 + 100$

Step 4.2

Add $0$ and $100$ .

$100$

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