Finite Math Examples

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

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

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

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

Step 1.1.1.2

Multiply

x

$x$ by

x

$x$ .

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

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

Step 1.1.1.3

Move

3

$3$ to the left of

x

$x$ .

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

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

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

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

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

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

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

$3 x$ from both sides of the equation.

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

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

Step 1.2.2

Subtract

23

$23$ from both sides of the equation.

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

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

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

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

Step 1.3

Simplify

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

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

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Step 1.3.1

Combine the opposite terms in

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

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

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Step 1.3.1.1

Subtract

3 x

$3 x$ from

3 x

$3 x$ .

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

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

Step 1.3.1.2

Add

x^{2}

$x^{2}$ and

0

$0$ .

x^{2} - 2 - 23 = 0

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

x^{2} - 2 - 23 = 0

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

Step 1.3.2

Subtract

23

$23$ from

- 2

$- 2$ .

x^{2} - 25 = 0

$x^{2} - 25 = 0$

x^{2} - 25 = 0

$x^{2} - 25 = 0$

x^{2} - 25 = 0

$x^{2} - 25 = 0$

Step 2

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

b^{2} - 4 (a c)

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

Step 3

Substitute in the values of

a

$a$ ,

b

$b$ , and

c

$c$ .

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

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

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

$0$ to any positive power yields

0

$0$ .

0 - 4 (1 \cdot - 25)

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

Step 4.1.2

Multiply

- 4 (1 \cdot - 25)

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

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Step 4.1.2.1

Multiply

- 25

$- 25$ by

1

$1$ .

0 - 4 \cdot - 25

$0 - 4 \cdot - 25$

Step 4.1.2.2

Multiply

- 4

$- 4$ by

- 25

$- 25$ .

0 + 100

$0 + 100$

0 + 100

$0 + 100$

0 + 100

$0 + 100$

Step 4.2

Add

0

$0$ and

100

$100$ .

100

$100$

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)

$(Δ)$ :

Δ > 0

$Δ > 0$ means there are

2

$2$ distinct real roots.

Δ = 0

$Δ = 0$ means there are

2

$2$ equal real roots, or

1

$1$ distinct real root.

Δ < 0

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

2

$2$ complex roots.

Since the discriminant is greater than

0

$0$ , there are two real roots.

Two Real Roots