a. find the value of the discriminant
b. describe the number and type of root
c. find the exact solution by using quadratic formula
1. p^2+12p=-4
2. 9x^2-6x+1=0
3. 2x^2-7x-4=0
2. 9x^2-6x+1=0
b^2-4ac=-6^2-4(9)(1) or 0
so it has 1 rational root.
p^2 + 12 p + 4 = 0
b^2-4ac = 144 -16 = 128
positive
so there will be two real roots
[ -12 +/- sqrt(128) ] / 2
2.
36 -4(9) = 36 -36 = 0
so
repeated real root
[ 6 +/- 0 ] / 18 = 1/3 and 1/3 (same, parabola just touches the x axis)
To answer each question about the given quadratic equations, we will need to follow a few steps. Let's break it down for each equation:
1. p^2 + 12p = -4
a. To find the value of the discriminant, we need to determine the coefficients of the quadratic equation - in this case, a, b, and c in the form ax^2 + bx + c = 0. In this equation, a = 1, b = 12, and c = -4. The formula for the discriminant is Δ = b^2 - 4ac. Plugging the values, we have Δ = (12)^2 - 4(1)(-4). Simplifying, we get Δ = 144 + 16 = 160.
b. To describe the number and type of roots, we look at the discriminant. If Δ > 0, there are two distinct real roots. If Δ = 0, there is one real root (which is also called a double root). If Δ < 0, there are no real roots, meaning there are two complex (conjugate pair) roots. In this case, Δ = 160 > 0, so there are two distinct real roots.
c. To find the exact solutions using the quadratic formula, we can use the formula: x = (-b ± √Δ) / 2a. Plugging in the values a = 1, b = 12, and c = -4, we have x = (-12 ± √160) / (2*1). Simplifying further, we get x = (-12 ± √160) / 2, which can be expressed as x = -6 ± 4√10.
2. 9x^2 - 6x + 1 = 0
a. The coefficients of this quadratic equation are a = 9, b = -6, and c = 1. Plugging these values in the discriminant formula, we have Δ = (-6)^2 - 4(9)(1) = 36 - 36 = 0.
b. Since Δ = 0, there is one real root.
c. Using the quadratic formula, x = (-b ± √Δ) / 2a. Substituting the values, we get x = (-(-6) ± √0) / (2*9), which simplifies to x = 6 / 18 = 1/3. Therefore, the exact solution is x = 1/3.
3. 2x^2 - 7x - 4 = 0
a. Here, a = 2, b = -7, and c = -4. Plugging these values into the discriminant formula, Δ = (-7)^2 - 4(2)(-4) = 49 + 32 = 81.
b. As Δ = 81 > 0, there are two distinct real roots.
c. Applying the quadratic formula, x = (-(-7) ± √81) / (2*2), which simplifies to x = (7 ± 9) / 4. This gives us two solutions: x = 16/4 = 4 and x = -2/4 = -1/2. Thus, the exact solutions are x = 4 and x = -1/2.