Quadratic Equation Standard Form

Which of the following systems of equations could a student use to write a quadratic function in standard form for the parabola passing through the points (-8,3), (-11,-6), and (-10,-1).

Which of the following are Quadratic equations?

• All

• (b) and (d)

• (a), (b) and (c)

• (a) and (c)

Answer explanation

(a) and (c) are quadratic polynomials whereas (b) and (d) are quadratic equations

Which of the following are quadratic equations? (a) (x ─ 1)² + 2 = 5x + 3 (b) x(5x + 3) =x² ─ 1

• Both (a) and (b)

• Only (a)

• Only (b)

• Neither

Answer explanation

(a) (x ─ 1)² + 2 = 5x + 3 x² ─ 2x + 1 + 2 ─ 5x ─ 3 =0 x² ─ 7x = 0 ---- Is a quadratic equation . (b) x(5x+3) =x² ─ 1 5x²+ 3x ─x²+1 = 0 4x² + 3x +1 = 0 ------Is a quadratic equation So the correct option is Option1.

The product of two consecutive integers is 168. Which of the following quadratic equations correctly represents this situation?

• x² + 2x - 168 =0

• x² + x + 168 =0

• x² + x - 168 =0

Answer explanation

Let the two consecutive integers be x, x+1 x ( x+1) = 168 x² + x = 168 x²+ x - 168 = 0 So the correct option is option 3. If Option 2 is chosen, then it is incorrect as the sign of 168 was not changed while transposing it to the LHS. If Option 1 is chosen, then it is also incorrect as instead of two consecutive integers x and x+1, x and x+2 were considered while formulating the equation.

The length of a rectangle is 3 units more than its width. If the width is represented by x and the area of the rectangle is 54 square meters, which of the following quadratic equations correctly represents this situation?

• x² + 3x +54 = 0

• x² + 3x ─ 54 = 0

• 3x² + x ─ 54 = 0

Answer explanation

Let the width of rectangle = x Then, length = x + 3 Area of the rectangle = length × width x (x +3) =54 x² + 3x ─ 54 =0 So the correct option is Option 2 If Option 1 is chosen, then it is incorrect as the sign of 54 was not changed while transposing it to the LHS. If Option 3 is chosen, then it is also incorrect as wrong multiplication of x(x+3) was done.

Which of the following are NOT quadratic equations? (a) x(x +2) = (x + 3)(x ─ 6) (b) (x + 1)³ = x³ ─ 6

• Both (a) and (b)

• Only (a)

• Only (b)

• Neither

Answer explanation

(a) x(x +2) = (x + 3)(x ─ 6) x² + 2x = x² ─ 6x + 3x ─ 18 2x = ─ 3x ─ 18 5x + 18 = 0 ---- This is a linear equation and NOT a quadratic equation (b) (x + 1)³ = x³ ─ 6 x³ + 3x² +3x + 1 = x³ ─ 6 3x² + 3x + 7 = 0 ---- This is a quadratic equation. So the correct option is Option 2.

If Farheen was younger by 5 years than her current age, then the square of her age would have been 11 more than 5 times her present age. If her present age is x, then which of the following quadratic equations correctly represents this situation?

• (x ─5 )² = 5x +11

• x ² = 5x +11

• (x ─5 )² + 11 = 5x

Answer explanation

Let Farheen's present age = x Then her age 5 years before = x ─ 5 Square of her age 5 years before = (x ─5 )² According to the question, the equation will be, (x ─5 )² = 5x +11 So the correct option is Option 1 If Option 2 is chosen, then it is incorrect. The student did not consider Farsheen's age 5 years ago but took the square of her present age in LHS. If Option 3 is chosen, then it is also incorrect. The student must have added 11 to the square of (x - 5) instead of 5x.

A particular quadratic function is known to have (8, 12) as its vertex coordinates and (0, -116) as its y-intercept. You are required to use this information to create

a). Its vertex equation;

b). Its standard equation.

Write your answers in the spaces provided below.

Determine a) the vertex-form quadratic function and b). the standard quadratic function using information shown on the following parabola (the graph of a quadratic function). Write your answers in the spaces provided.

A parabolic graph that is known to model (represent) a quadratic function has (-2, -7) as its vertex coordinates and (0, -3) as its y-intercept. Determine

a). Its vertex equation

b). Its Standard equation.

Write your answers in the spaces provided below.