The Quadratic Formula (Math Masters)

What is the quadratic formula?

• x=(-a±√(b²-4ac))/(2b)

• x=(-b±√(b²-4ac))/(2a)

• x=(b±√(b²+4ac))/(2a)

• x=(b²-4ac)/(2a)

Answer explanation

The quadratic formula is used to find the roots of a quadratic equation. The correct formula is x=(-b±√(b²-4ac))/(2a), which allows for calculating the values of x based on coefficients a, b, and c.

What does the term b²-4ac represent in the quadratic formula?

• The vertex of a quadratic equation.

• The slope of a linear equation.

• The root of a polynomial function.

• The discriminant of a quadratic equation.

Answer explanation

The term b²-4ac is known as the discriminant of a quadratic equation. It determines the nature of the roots: if positive, there are two distinct real roots; if zero, one real root; if negative, no real roots.

If the discriminant is positive, how many real solutions does the quadratic equation have?

• 1 real solution

• No real solutions

• Infinite real solutions

• 2 real solutions

Answer explanation

If the discriminant of a quadratic equation is positive, it indicates that there are two distinct real solutions. Therefore, the correct answer is 2 real solutions.

For the quadratic equation 2x²+4x+2=0, what are the values for a,b, and c?

• a = 2, b = 4, c = 2

• a = 3, b = 4, c = 1

• a = 2, b = 2, c = 4

• a = 1, b = 2, c = 3

Answer explanation

In the quadratic equation 2x² + 4x + 2 = 0, the coefficients are identified as a = 2, b = 4, and c = 2. Therefore, the correct choice is a = 2, b = 4, c = 2.

What is the discriminant for the equation x²-6x+9=0?

• 0

• -1

• 4

• 6

Answer explanation

The discriminant is calculated using the formula b² - 4ac. For the equation x² - 6x + 9, a = 1, b = -6, and c = 9. Thus, the discriminant is (-6)² - 4(1)(9) = 36 - 36 = 0. Therefore, the correct answer is 0.

How many real solutions does the equation x²+4x+5=0 have?

• 1

• 2

• 3

• None

Answer explanation

The equation x² + 4x + 5 can be analyzed using the discriminant (b² - 4ac). Here, a=1, b=4, c=5. The discriminant is 4² - 4(1)(5) = 16 - 20 = -4, which is negative. Therefore, there are no real solutions.

Using the quadratic formula, what are the solutions for x²-2x-3=0?

• x = 3, x = -1

• x = 0, x = 4

• x = 1, x = 3

• x = 2, x = -3

Answer explanation

To solve x² - 2x - 3 = 0 using the quadratic formula x = (-b ± √(b² - 4ac)) / 2a, where a=1, b=-2, c=-3. This gives x = (2 ± √(4 + 12)) / 2 = (2 ± 4) / 2. Thus, x = 3 and x = -1 are the solutions.

What will the value of the discriminant be for the quadratic equation x²+2x+1=0?

• 1

• 4

• -1

• 0

Answer explanation

The discriminant of a quadratic equation ax² + bx + c is given by b² - 4ac. For x² + 2x + 1, a=1, b=2, c=1. Thus, the discriminant is 2² - 4(1)(1) = 4 - 4 = 0. Therefore, the value of the discriminant is 0.

What is the first step in applying the quadratic formula to solve 3x²-12x+9=0?

• Calculate the discriminant.

• Factor the quadratic equation.

• Identify the coefficients a, b, and c.

• Graph the quadratic function.

Answer explanation

The first step in applying the quadratic formula is to identify the coefficients a, b, and c from the equation 3x² - 12x + 9 = 0. Here, a = 3, b = -12, and c = 9, which are essential for using the formula.

If the quadratic equation has a discriminant of 25, what can we say about the solutions?

• The quadratic equation has two complex solutions.

• The quadratic equation has no real solutions.

• The quadratic equation has two distinct real solutions.

• The quadratic equation has one real solution.

Answer explanation

A discriminant of 25 indicates that the quadratic equation has two distinct real solutions, as it is positive. This means the parabola intersects the x-axis at two points.