Math Olympiad Test: Quadratic Equations- 2 - Class 10 MCQ

Simplifying x² + -6 = 138
Reorder the terms: -6 + x² = 138
Solving -6 + x² = 138
Solving for variable 'x'
Move all terms containing x to the left, all other terms to the right
Add '6' to each side of the equation
-6 + 6 + x² = 138 + 6
Combine like terms: -6 + 6 = 0
0 + x² = 138 + 6
x² = 138 + 6
Combine like terms: 138 + 6 = 144
x² = 144
Simplifying x² = 144
Take the square root of each side:
x = {-12, 12}

This equation is written in standard form and it is set equal to 0. Therefore, I know that the 2 methods for solving are factoring and using the quadratic equation.
As I browse the answers, it looks like the values for x are integers, so I must be able to factor. Therefore, this will be the easiest method.
I need to find two integers whose product is -32 and whose sum is 4. (8 & -4).
Don’t make the mistake of assuming these are your answers!!!
(x + 8) (x - 4) = 0 Rewrite as factors
x + 8 = 0 x - 4 = 0
Set each factor equal to 0
x + 8 - 8 = 0 - 8   
x - 4 + 4 = 0 + 4
x = - 8,  x = 4

In order to find the values for y, we must set the equation equal to 0 by subtracting 5 from both sides.
2y² - 5y + 2 - 5 = 5 - 5
Subtract 5 from both sides
2y² - 5y - 3 = 0
Now the equation is set equal to 0.
As I take a look at the answers, it looks like this equation can be factored. Let’s try!
(2y + 1) (y - 3) = 0 Factor the equation
2y + 1 = 0   y - 3 = 0
Set each factor equal to 0
2y + 1 - 1 = 0 - 1     y - 3 + 3 = 0 + 3
2y = -1   y = 3

In order to find the maximum height of the ball, I would need to find the y-coordinate of the vertex.
Vertex formula: x = - b/2a
where: a = -16 b = 36 c = 1.5

x = 1.125
Now substitute 1.125 for t into the equation and solve for h(t). H(t) = -16 (1.125)² + 36 (1.125) + 1.5
H(t) = 21.75

The graph of a quadratic equation is parabola.
If a < 0, then the parabola opens downward. If the parabola opens downward, then the vertex is the point whose y-value is the maximum value of f.

Quadratic function is a function that can be described by an equation of the form fxx = ax² + bx + c, where a ≠ 0.
In a quadratic function, the greatest power of the variable is 2. The graph of a quadratic function is a parabola.

Given equation is
3x² + 8x + 2 = 0
and α, β are its roots then

Here Kx² + 2x + 3K = 0
then given Sum of roots = Product of roots

The required equation is 
x² - (Sum of roots)x + Product of roots = 0
⇒ x² - [7 + (- 3)]x + (7) (- 3) = 0
⇒ x² - 4x - 21 = 0

Here x² + px + q = 0 has one root a = 4
∴ 42 + b⁴ + 12 = 0
⇒ b = -7
Now the equation x² - 7x + q = 0 has equal roots
(-7)² - 4 (1) (q) = 0 then
⇒ 4q = 49 ⇒ q = 49/4

The given equation is
x² - p(x + 1) - c = 0
⇒ x² - px + (-c - p) = 0
∴ α + β = p
and

Now (α + 1) (β+ 1) = αβ + α + β + 1 = - c -p + p + 1 = 1 - c

We have x² - px + q = 0
Now α + β = p, αβ = q
∴ α - β = 1
2α = p + 1

Now
β = p - α

and     αβ = q

p² - 1 = 4q ⇒ p² - 4q = 1

Given a and b are roots of x² + x + 1 = 0
then a + b = -1 and ab = 1
∴ a² + b² = (a + b)² - 2ab
= (-1)² - 2(1) = 1 - 2 = -1

Given A and B are the roots of x² - 12x + 27 = 0
⇒ (x - 3)(x - 9) = 0
∴ A =  3, B = 9
Hence A³ + B³ = 3³ + 9³ = 27 + 729 = 756

If the quadratic equation is given by Ax² + Bx + C = 0.
Then,

Given: (a² - b²)x² + (b² - c²)x + (c² - a²) = 0
Where,
A = (a² - b²), B = (b² - c²), C = c² - a²
Calculation:
Here,
A + B + C = (a² - b²) + (b² - c²) + (c² - a²) = 0.
So, we can say that x = 1 is a root of a given quadratic equation. According to question both roots are same.
Then,
Roots (x) = 1, 1


(a² - b²) = (c² - a²)
By arranging, we get
b² + c² = 2a²