Question

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Answer 1

1- c

Solve Using the Quadratic Formula 9x^2-24x+16=0

9x2−24x+16=09x2-24x+16=0

Use the quadratic formula to find the solutions.

−b±√b2−4(ac)2a-b±b2-4(ac)2a

Substitute the values a=9a=9, b=−24b=-24, and c=16c=16 into the quadratic formula and solve for xx.

24±√(−24)2−4⋅(9⋅16)2⋅924±(-24)2-4⋅(9⋅16)2⋅9

Simplify.


Simplify the numerator.

x=24±02⋅9x=24±02⋅9

Simplify the denominator.

x=24±018x=24±018

Simplify 24±01824±018.

x=2418x=2418

Reduce the expression by cancelling the common factors.



Factor 66 out of 1818.

x=6⋅46⋅3x=6⋅46⋅3

Cancel the common factor.

x=6⋅46⋅3x=6⋅46⋅3

Rewrite the expression.

x=43x=43

The final answer is the combination of both solutions.

x=43x=43 Double roots


1-d
Find the factors of the quadratic trinomial first.

3x2−5x+2=0

(3x−2)(x−1)=0

Solve each factor equal to 0.

3x−2=0 →3x=2 →x=23

x−1=0 →x=1

Of the two solutions, x=1 is the larger.

2-A

 d will be > 0 for the equation to have real and distinct roots...

a = k, b = 4, c = 1

b2 - 4ac > 0

16 - 4(k)(1) > 0

16 - 4k > 0

4k<16

k < 4

so k can have any value less than 4

2-B
Given kx2 – 2√5x + 4 = 0 has equal roots 
Hence b2 – 4ac = 0
That is (– 2√5)2 – 4(k)(4) = 0
20 – 16k = 0
∴ k = 5/4

2-C
The given equation 3x2-5x+2k=0 is in the form of ax2+bx+c=0 where a= 3, b= -5, c= 2k Given that, the equation has real and equal roots D= b2-4ac=0 = (-5)2-4(3)(2k)=0 = 25-24k =0 K = k=2524 The value of the k is k=2524 

2-D
Simplifying 4x2 + kx + 9 = 0 Reorder the terms: 9 + kx + 4x2 = 0 Solving 9 + kx + 4x2 = 0 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + kx + -9 + 4x2 = 0 + -9 Reorder the terms: 9 + -9 + kx + 4x2 = 0 + -9 Combine like terms: 9 + -9 = 0 0 + kx + 4x2 = 0 + -9 kx + 4x2 = 0 + -9 Combine like terms: 0 + -9 = -9 kx + 4x2 = -9 Add '-4x2' to each side of the equation. kx + 4x2 + -4x2 = -9 + -4x2 Combine like terms: 4x2 + -4x2 = 0 kx + 0 = -9 + -4x2 kx = -9 + -4x2 Divide each side by 'x'. k = -9x-1 + -4x Simplifying k = -9x-1 + -4x

2-F

the equal roots means the dicriminant will be equal to 0.
that is, b2 - 4 ac =0

here, a=9, b=-24 and c=k

hence, (-24)2 - 4 * 9 * k=0
 576 - 36k =0
576=36k
k= 576 /36
k= 16

now,
9x2 - 24x + 16=0
9x2 - 12x -12x +16=0
3x(3x-4) - 4  (3x-4)
3x-4  or  3x-4

therefore roots are:
4/3 and 4/3

2-G
D=b^2-4ac
a=1
b=-2(k+1)
c=k^2
=-2(k+1)^2-4k^2
=(-2k-2)^2-4k^2
=[(4k^2+4+2(-2k)(-2)]-4k^2
=8k+4=0
k=-1/2