2015 MTAP Reviewer for Grade 9 Solutions(21-25)

This is already the 5th part of the reviewer solution series for grade 9 MMC. You may check the previous parts in this section.

Problem 21:

Compute the sum of all the roots of (x-2)(x+1)+(x-1)(x+4)=0

Solution:

To find the sum of the roots, we expand the equation,

(x-2)(x+1)+(x-1)(x+4)=0

x^2-x-2+x^2+3x-4=0

2x^2+2x-6=0

x^2+x-3=0

Now, the sum of the roots of this quadratic equation is just the negative ratio of b and a. Thus, the sum is -1.

Problem 22:

If r and s are the roots of x^2+x-1=0, evaluate (r+s)^2

Solution:

The sum of the roots of quadratic equation is again the negative ratio of b and a. Thus,

r+s=-1

This means that \boxed{(r+s)^2=1}.

Actually, this problem doesn’t really make sense at all. I was expecting a twist somewhere. They might mistyped this (r+s)^2 from this (r-s)^2. The latter is more challenging and adds flavor to the problem. Anyways, let’s proceed.

Problem 23:

For what value(s) of m are the roots of (m-1)x^2-mx+1=0 equal?

Solution:

For a quadratic equation to have an equal roots, the discriminant must be equal to 0. Thus,

D=0

b^2-4ac=0

(-m)^2-4(m-1)(1)=0

m^2-4m+4=0

(m-2)(m-2)=0

\boxed{m=2}

Problem 24:

It is known that y varies as the square of x and y=8 when x=1. What is y when x=8?

Solution:

Let k be the constant of proportionality. Since y varies as the square of x we have,

y=kx^2

Solving for k when  y=8 and x=1,

8=k(1)^2

k=8

Solving for y when x=8,

y=kx^2

y=8(8)^2

y=8(64)

y=8(64)

\boxed{y=512}

Problem 25:

Suppose that x and y are inversely proportional and are positive quantities. By what percent does y decrease if x is increased by 25%?

Solution:

We let k be the constant of proportionality and we establish the equation.

y=\dfrac{k}{x}

IF x in increased by 25%, the value of x will become 1.25x. Since the constant of proportionality is constant. We equate the ks of two situations.

k_1=k_2

x_1y_1=y_2x_2

x_1y_1=y_2(1.25x_1)

y_2=\dfrac{x_1y_1}{1.25x_1}

y_2=\dfrac{y_1}{1.25}

y_2=0.8y_1

From 1, the value of y decreases to 0.8. Thus, it decreased by 20%.

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