Chapter 4 - Equations

the sum and products of the roots

If a𝑥² + b𝑥 + c = 0 has roots α and β, then α + β = -b/a and αβ = c/a

the null factor law

If the product of two (or more) numbers is zero, then at least one of them must be zero. So, if ab = 0 then a = 0 or b = 0.

The roots of the equation 4𝑥² + 5𝑥 - 1 = 0 are α and β. Find a quadratic equation with roots 3α and 3β.

If α and β are the roots of 4𝑥² + 5𝑥 - 1 = 0, then α + β = -5/4 and αβ = -1/4

Find the sum and product of the roots of 25𝑥² - 20𝑥 + 1 = 0. Check your answer by solving the quadratic.

If α and β are the roots, then -b/a = 20/25 = 4/5 and αβ = c/a = 1/25

Solve for 𝑥 using the null factor law: 3𝑥(𝑥 - 5) = 0

3𝑥(𝑥 - 5) = 0 3𝑥 = 0 or 𝑥 - 5 = 0 ∴ 𝑥 = 0 or 5

factored form

an equation is written in factored form if one side is fully factorized and the other side is zero

quadratic equation

an equation of the form ax² + bx + c = 0, where a ≠ 0

completing the square

used when quadratic cannot be factored easily ex: 𝑥² - 10𝑥 + 21 = 0

The discriminant of a quadratic

- If Δ > 0, √Δ is a positive real number, so there are two distinct real roots 𝑥 = (-b ± √Δ) / 2a and 𝑥 = (-b - √Δ) / 2a - If Δ = 0, 𝑥 = -b / 2a is the only solution, which we call a repeated root. - If Δ < 0, √Δ is not a real number so there are no real roots. - If a, b, and c are rational and Δ is a square, then the equation has two rational roots which can be found by factorization.

ways to solve quadratic equations

- rewrite the quadratic into factored form then use null factor law - rewrite the quadratic into completed square form then solve (𝑥 - h)² = k - use the quadratic formula - use technology

solving by factorization

Step 1: If necessary, rearrange the equation so one side is zero. Step 2: Fully factorize the other side. Step 3: Use the null factor law. Step 4: Solve the resulting linear equations.

Solve by completing the square: 𝑥² - 10𝑥 + 21 = 0

Step 1: Rearrange the formula so that the constants are on the right hand side. 𝑥² - 10𝑥 = -21 Step 2: Add (b/2)² to both sides of the equation 𝑥² - 10𝑥 + (-10/2)² = -21 + (-10/2)² ∴ 𝑥² - 10𝑥 + 25 = -21 + 25 Step 3: Factor the left hand side (𝑥 - 5)² = 4 Step 4: Solve (𝑥 - 5)² = 4 𝑥 - 5 = ±√4 𝑥 - 5 = ± 2 𝑥 = 5 ± 2 𝑥 = 7 or 3

solving polynomial equations using technology (TI-84)

To solve polynomial equations on the calculator, we use the polynomial root finder. When using the polynomial root finder, the equation must be in polynomial form. To solve 2𝑥² + 4𝑥 = 7, we rearrange the equation to get 2𝑥² + 4𝑥 - 7 = 0. To access the polynomial root finder, press APPS, select 9: PlySmlt2, select 1: POLY ROOT FINDER, then set up the screen as shown. To select an option, highlight it, the press ENTER. Now press GRAPH (NEXT), and enter the quadratic as shown. Press GRAPH (SOLVE) to solve the equation. So 𝑥 ≈ 1.12 or -3.12

power equation

an equation which can be written in the form 𝑥ⁿ = k, n ≠ 0

Δ (discriminant)

b² - 4ac

Use the discriminant to determine the nature of the roots of 3𝑥² - 4𝑥 - 2 = 0

Δ = b^2 - 4ac = (-4)^2 - 4(3)(-2) = 40 Since Δ > 0, but 40 is not a square, there are 2 distinct irrational roots.

Use the discriminant to determine the nature of the roots of 2𝑥² - 2𝑥 + 3 = 0

Δ = b² - 4ac = (-2)² - 4(2)(3) = -20 Since Δ < 0, there are no real roots.

Solve (𝑥 - 4)(3𝑥 + 7) = 0 using the null factor law.

𝑥 - 4 = 0 or 3𝑥 + 7 = 0 𝑥 = 4 or 3𝑥 = -7 ∴ 𝑥 = 4 or -7/3

Consider 𝑥² - 2𝑥 + m = 0. Find the discriminant Δ, and hence find the values of m for which the equation has: a) a repeated root, b) 2 distinct real roots, c) no real roots.

𝑥² - 2𝑥 + m = 0 has a = 1, b = -2, c = m Δ = b² - 4ac = (-2)² - 4(1)(m) = 4 - 4m a) For a repeated root Δ = 0 4 - 4m = 0 m = 1 b) For 2 distinct real roots Δ > 0 4 - 4m > 0 -4m > -4 m < 1 c) For no real roots Δ < 0 4 - 4m < 0 -4m < -4 m > 1