Quadratic Equations

Which statement best summarizes the chapter "Quadratic Equations"?

1. Quadratic equations always have one solution.
2. Quadratic equations can only be solved by factorization.
3. Quadratic equations are second-degree equations that can be solved using factorization, completing the square, or the quadratic formula, and the discriminant determines the nature of their roots.
4. Quadratic equations always have imaginary roots.

Which graph represents a quadratic equation?

1. Straight line
2. Circle
3. Parabola
4. Rectangle

Which statement about quadratic equations is correct?

1. Every quadratic equation has exactly one root.
2. Every quadratic equation has degree 2.
3. Every quadratic equation is linear.
4. Quadratic equations never have equal roots.

Which equation has equal roots?

1. x^2 - 4 = 0
2. x^2 + 6x + 9 = 0
3. x^2 - 5x + 6 = 0
4. x^2 + 1 = 0

The product of the roots of ax^2 + bx + c = 0 is:

1. -b/a
2. c/a
3. a/c
4. b/c

The equation x^2 + 4x + 4 = 0 has:

1. Two distinct roots
2. Equal roots
3. No real roots
4. Three roots

Which method is NOT commonly used to solve quadratic equations?

1. Factorization
2. Completing the square
3. Quadratic formula
4. Long division only

If the discriminant is positive, the equation has:

1. Equal roots
2. No real roots
3. Two distinct real roots
4. One root only

The roots of x^2 - 9 = 0 are:

1. 3 and -3
2. 9 and 1
3. 0 and 9
4. -9 and 3

The quadratic formula is:

1. (-b plus-or-minus sqrt(b^2-4ac)) / 2a
2. (b plus-or-minus sqrt(4ac)) / a
3. (-a plus-or-minus sqrt(b^2)) / 2c
4. 2a / (b+c)

If the discriminant of a quadratic equation is zero, then the roots are:

1. Distinct real roots
2. Equal real roots
3. Imaginary roots
4. Irrational roots only

The discriminant of the quadratic equation ax^2 + bx + c = 0 is:

1. b^2 + 4ac
2. 4ac - b^2
3. b^2 - 4ac
4. 2ac - b

The roots of the equation x^2 - 5x + 6 = 0 are:

1. 2 and 3
2. 1 and 6
3. -2 and -3
4. 0 and 6

What is the standard form of a quadratic equation?

1. ax + b = 0
2. ax^2 + bx + c = 0, where a is not equal to 0
3. ax^3 + bx + c = 0
4. a + bx = c

The graph of a quadratic equation is always a:

1. Straight line
2. Circle
3. Parabola
4. Triangle

The quadratic equation whose roots are 2 and −3 is:

1. x² − x − 6 = 0
2. x² + x − 6 = 0
3. x² − x + 6 = 0
4. x² + 5x − 6 = 0

If one root of a quadratic equation is 3 and sum of roots is 8, the other root is:

1. 4
2. 5
3. 6
4. 7

Which of the following equations has no real roots?

1. x² − 4x + 4 = 0
2. x² + 1 = 0
3. x² − 5x + 6 = 0
4. x² − x = 0

If roots of a quadratic equation are equal, then discriminant is:

1. Positive
2. Negative
3. Zero
4. One

Product of roots of 2x² + 5x + 3 = 0 is:

1. 5/2
2. 3/2
3. 3
4. 2

The sum of roots of x² − 7x + 12 = 0 is:

1. 7
2. −7
3. 12
4. −12

If sum of roots is 0, then:

1. b = 0
2. c = 0
3. a = 0
4. D = 0

A quadratic equation can have maximum how many real roots?

1. 1
2. 2
3. 3
4. Infinite

If x² − 4 = 0, roots are:

1. 4, −4
2. 2, −2
3. 0, 4
4. 1, −1

Quadratic formula is used to find:

1. Only factors
2. Only roots
3. HCF
4. Graph slope

Nature of roots of x² + x + 1 = 0 is:

1. Real and distinct
2. Real and equal
3. Imaginary
4. Rational

If discriminant is 0, then roots are:

1. Real and distinct
2. Real and equal
3. Imaginary
4. Cannot be determined

The roots of x² − 5x + 6 = 0 are:

1. 1, 6
2. 2, 3
3. −2, −3
4. 5, 6

Which of the following is a quadratic equation?

1. 2x + 3 = 0
2. x² + 3x + 2 = 0
3. x³ − x = 0
4. 1/x + x = 0

If a,b are roots of x²–6x+6=0, value of a²+b² is

1. 36
2. 24
3. 12
4. 6

Product of roots of ax–ax²=1 is

1. 1/a²
2. a
3. 1
4. 1/a

Number of roots satisfying √(5–x)=x·√(5–x) is

1. 1
2. 2
3. 3
4. 7

If √(x–1)–√(x+1)=1/√(4x+1), then 4x equals

1. 5/4
2. 0
3. 5
4. 1

Sum of reciprocals of roots of x²+px+q=0 is

1. q/p
2. –p/q
3. p/q
4. –q/p

If discriminant of an equation with a, c real constants is zero, roots must be

1. equal and integral
2. rational and equal
3. real and equal
4. imaginary

Roots of ax²+bx+c=0 are reciprocal if

1. a=b
2. a=b=c
3. c=a
4. b=c

Roots of the equation where D=0 are

1. real and equal
2. rational and equal
3. rational and unequal
4. imaginary

If a,b are roots of x²–px+q=0, then a²+b² =

1. p²+q²
2. p²+2q
3. p²–q²
4. p²–2q

Values of k for which 2x²–kx+x+8=0 has real and equal roots are

1. 9 and –7
2. only 9
3. only –7
4. –9 and 7

If one root of a(b–c)x²+b(c–a)x+c(a–b)=0 is 1, the other root is

1. b(c–a)/a(b–c)
2. c(a–b)/a(b–c)
3. a(b–c)/c(a–b)
4. c(a–b)/a(b–c)

Difference of roots of x²–7x–9=0 is

1. 7
2. –9
3. √85
4. 7/√85

Roots of mx(x–1)+1=0 are equal when

1. m=1
2. m=–1/2
3. m=1
4. m=7

The equation √(x–2)=4 has

1. two real roots and one imaginary
2. one real and one imaginary
3. two imaginary roots
4. one real root

Least value of ax²+bx+c (a>0) is

1. 4ac–b²/4a
2. (4ac–b²)/4a
3. b²–4ac/4a
4. 4ac+b²/4a

Perimeter of rectangle is 34m and diagonal is 13m. Dimensions are

1. 9m & 2m
2. 15m & 2m
3. 12m & 5m
4. 17m & 7m

Product of two consecutive odd numbers is 143. The numbers are

1. 11 & 13
2. 13 & 15
3. –11 & 13
4. –11 & 15

Value of k if x²+kx+1 is factorizable into two linear factors

1. k≥2
2. k≤–2
3. k≥2 or k≤–2
4. neither

If roots of x²–px+q=0 differ by unity, then

1. p²=4q+1
2. p²=4q–1
3. p²=4p+1
4. p²=4p–1

Sum of roots of 3x²+(2k+1)x–(k+5)=0 equals product of roots. Value of k is

1. –4
2. 4
3. 1
4. –1

If a=b=c, roots of (x–a)(x–b)+(x–b)(x–c)+(x–c)(x–a)=0 are

1. real and unequal
2. imaginary
3. real and equal
4. imaginary and equal

If roots of (a–b)x²+(b–c)x+(c–a)=0 are equal, then

1. 2b=a+c
2. 2a=b+c
3. 2c=a+b
4. a=b+c

If one root of x²+ax+4=0 is twice the other, the value of 'a' is

1. ±3√2
2. ±2√3
3. ±4√2
4. 7

Coefficient of x was taken as 17 instead of 13 in x²+px+q=0; roots were –2 and –15. Original roots are

1. –3 or 10
2. –3 or –10
3. 3 or –10
4. 3 or 10

If α, β are roots of x²+px+12=0 and α–β=1, then p =

1. 1
2. 7 or 1
3. –7
4. 7 or –7

If x=√(1+√(1+√(1+...))), then x =

1. (1+√5)/2
2. (1–√5)/2
3. (√5–1)/2
4. (√5+1)/2

If a root of ax²+bx+c=0 is reciprocal of a root of ax²+bx+c=0, then

1. (cc–aa)²=(ba–cb)(ab–bc)
2. (bb–aa)²=(ca–bc)(ab–bc)
3. (cc–aa)²=(ba+cb)(ab+bc)
4. (c–c)²=(a–a)²

Rectangle perimeter is 82m, area 400m². Length of diagonal is

1. √882
2. √881
3. √869
4. √892

Out of swans, 7/2 times square root of total are on bank, two are in water. Total swans =

1. 16
2. 17
3. 18
4. 19

Value of a such that x=3 is a solution of ax²+2x+3=0 is

1. 0
2. 1
3. 2
4. –1

If c≠0 and p/(x+c)+p/(x–c)=(2a+2b)/x has two equal roots, then p =

1. a+b±2√(ab)
2. (a+b)±√(ab)
3. a+b
4. a–b

The roots of (x²–3x+2)(x)(x–4)=0 are

1. 1, 3 and 4
2. 0 and 4
3. 0, 1, 2 and 4
4. 1, 2 and 4

One year ago a man was 8 times as old as his son. Now his age equals square of son's age. Present ages are

1. 7 years, 49 years
2. 8 years, 48 years
3. 9 years, 50 years
4. 10 years, 51 years

Values of k for which x²–4x+k=0 has distinct real roots is

1. k < 4
2. k > 4
3. k = 4
4. k is undefined

If –4 is a root of x²+px–4=0, and x²+px+k=0 has equal roots, then k =

1. 9/4
2. 4/9
3. 3/4
4. 9/4

Which equation has imaginary roots?

1. x²–4x+2=0
2. 3x²+2x–1=0
3. x²–4x+2=0
4. x²+x+1=0

Values of x satisfying 3/(2x²–5x+5) = 10/(x²–3x+4) are

1. 5, 5/2
2. 5, 5/2
3. 2, 5/2
4. 5, 3/2

If 2+√3 is a root of x²+px+q=0 (p,q real), then (p,q) =

1. (–4,7)
2. (4,–7)
3. (4,7)
4. (–4,–7)

Two numbers differ by 4 and their product is 192. The numbers are

1. 12, 16
2. 10, 14
3. 11, 15
4. 13, 18