Can someone show me how these 2 are done???

Question

1. Solve by using the quadratic formula:
4x^2 + 7x + 5 = 0

2. Solve:
x^2 - 4x - 21 = 0

Answer 1

1) 4x² + 7x + 5 = 0
From the general equation "ax²+bx+c=0", a formula hs been derived to find the value of "x"
x = {-b±√(b²-4ac)} /2a; where, a is the coeffient of x², b of x, and c,the constant.
Let us substitute the values.

x = { -7 ± √(7²- 4*4*5)} / 2*4
x = { - 7± √(49-80)} / 8
x = { - 7± √(-31) / 8}
Square root of a negative quantity can be neglected
x = {-7 / 8} = - (7/8).....Ans
===============

2) x² - 4x - 21 = 0
This can be solved by "splitting the middle term" method.
The middle term should be split up into two parts in such a way that their sum will be equal to the middle term an their product will be equal to the product of the two extreme terms.
ie. -4x = -7x +3x
Again, . . .(-7x) * (+3x) = -21x² (product of middle terms)
Similarly, (x²) * (-21) = -21x²..(product of extreme terms)
Hence the equation can be rewritten as,
. . . . . .. . x² - 7x + 3x - 21 = 0
. . . . . . . .x(x-7) +3(x-7) = 0
. . . . . . . . (x-7) (x+3) = 0
When the productof two numbers = 0, any one of the quantities will be 0.
Let (x-7) = 0
x = 7 . . . Ans(1)
Let (x+3) = 0
x = -3 . . .Ans(2)
The values of x are 7 or -3
. . . . . . . . . . . . . .. =======

Answer 2

The quadratic formula is (-b +/- sqrt(b^2-4ac))/2a.

a = the nbr in front of the x^2
b = the nbr in front of x
c = the constant (no x-term)

So for 1) a = 4, b = 7 & c = 5

[-7 +/- sqrt (7^2 - 4(4)(5) ] / 2(4)
= [-7 +/- sqrt (-31) ] / 8
= -7/8 +/- [sqrt(31)*i]/8 where i is defined as sqrt (-1)

Same logic again for the second question.


BTW - Sanyarem used the wrong formula to solve the quadratic: it's -b in the numerator not b^2 and it's 2a on the denominator not a^2.

Answer 3

just remember that: ax^2 + bx + c = 0 is standard form, and as #1 is already in standard form, you just have to plug in your numbers into the quad. formula as follows:

[-b^2 +/- sqrt(b^2-4ac)]/a^2
=[-7 +/- sqrt(49-4(4 X 5))]/16
=[-7 +/- sqrt(49-80)]
right here, you see a negative inside the square root, which means there is no x intercept, unless i made a mistake.

2.
x^2 - 4x - 21 = 0
factor
(x+3)(x-7)=0
x=-3
x=7
factoring just takes practice.

Answer 4

Question 1
x = [- 7 ± √(49 - 80) ] / 8
x = [ - 7 ± √(31) ] / 8 is exact answer.
√31 may be calculated to give approximate answer if required.

Question 2
(x - 7) (x + 3) = 0
x = 7 , x = - 3

Answer 5

its called a calculator