2) One of the roots of a quadratic equation is 4 - 77. Find the equation.
Answers
EXPLANATION.
One roots of the quadratic equation = 4 - 77.
As we know that,
If the quadratic equation is in the form of,
⇒ ax² + bx + c = 0 (a ≠ -1).
⇒ Their roots are = α,β.
⇒ If one roots = 4 - 77.
⇒ Other roots is their Conjugate roots pair = 4 + 77.
As we know that,
Sum of the zeroes of the quadratic equation.
⇒ α + β = -b/a.
⇒ 4 - 77 + 4 + 77.
⇒ 8.
⇒ α + β = 8.
Products of the zeroes of thee quadratic equation.
⇒ αβ = c/a.
⇒ (4 - 77)(4 + 77).
As we know that,
Formula of :
⇒ x² - y² = (x + y)(x - y).
Using this formula in equation, we get.
⇒ [(4)² - (77)²].
⇒ [16 - 5929].
⇒ -5913.
⇒ αβ = - 5913.
As we know that,
Formula of quadratic equation.
⇒ x² - (α + β)x + αβ.
Put the values in the equation, we get.
⇒ x² - (8)x + (-5913) = 0.
⇒ x² - 8x - 5913 = 0.
MORE INFORMATION.
Conjugate roots.
(1) = D < 0.
One roots = α + iβ.
Other roots = α - iβ.
(2) = D > 0.
One roots = α + √β.
Other roots = α - √β.
Step by Step Solution
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Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
x*(x-4)-(77)=0
Step by step solution :
STEP
1
:
Equation at the end of step 1
x • (x - 4) - 77 = 0
STEP
2
:
Trying to factor by splitting the middle term
2.1 Factoring x2-4x-77
The first term is, x2 its coefficient is 1 .
The middle term is, -4x its coefficient is -4 .
The last term, "the constant", is -77
Step-1 : Multiply the coefficient of the first term by the constant 1 • -77 = -77
Step-2 : Find two factors of -77 whose sum equals the coefficient of the middle term, which is -4 .
-77 + 1 = -76
-11 + 7 = -4 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -11 and 7
x2 - 11x + 7x - 77
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-11)
Add up the last 2 terms, pulling out common factors :
7 • (x-11)
Step-5 : Add up the four terms of step 4 :
(x+7) • (x-11)
Which is the desired factorization
Equation at the end of step
2
:
(x + 7) • (x - 11) = 0
STEP
3
:
Theory - Roots of a product
3.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Solving a Single Variable Equation:
3.2 Solve : x+7 = 0
Subtract 7 from both sides of the equation :
x = -7
Solving a Single Variable Equation:
3.3 Solve : x-11 = 0
Add 11 to both sides of the equation :
x = 11