1. Bob’s age is twice that of his sister. When you add Bob's age to his sister's age, you get 24. How old is Bob and his sister?
2 answers:
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Answer:
either x = -1, or x = - 9
Step-by-step explanation:
given
add one to both sides
im not sure which one comes first. either you multiply both the x and the 5 in the parenthesis by . or you square x and 5
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation : 1/4*(x+5)^2-(4)=0
Step by step solution :
Step 1 :
1 /4
Equation at the end of step 1 :
1
(— • (x + 5)2) - 4 = 0
4
Step 2 :
Equation at the end of step 2 :
(x + 5)2
———————— - 4 = 0
4
Step 3 :
Rewriting the whole as an Equivalent Fraction :
3.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using 4 as the denominator :
4 4 • 4
4 = — = ———
1 4
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
3.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
(x+5)2 - (4 • 4) x2 + 10x + 9
——————— ——————— = ——
4 4
Trying to factor by splitting the middle term
3.3 Factoring x2 + 10x + 9
The first term is, x2 its coefficient is 1 .
The middle term is, +10x its coefficient is 10 .
The last term, "the constant", is +9
Step-1 : Multiply the coefficient of the first term by the constant 1 • 9 = 9
Step-2 : Find two factors of 9 whose sum equals the coefficient of the middle term, which is 10 .
-9 + -1 = -10
-3 + -3 = -6
-1 + -9 = -10
1 + 9 = 10 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 1 and 9
x2 + 1x + 9x + 9
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x+1)
Add up the last 2 terms, pulling out common factors :
9 • (x+1)
Step-5 : Add up the four terms of step 4 :
(x+9) • (x+1)
Which is the desired factorization
Equation at the end of step 3 :
(x + 9) • (x + 1)
————————————————— = 0
4
Step 4 :
When a fraction equals zero :
4.1 When a fraction equals zero ...
Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.
Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.
Here's how:
(x+9)•(x+1)
——————————— • 4 = 0 • 4
4
Now, on the left hand side, the 4 cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
(x+9) • (x+1) = 0
Theory - Roots of a product :
4.2 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 :
4.3 Solve : x+9 = 0
Subtract 9 from both sides of the equation :
x = -9
Solving a Single Variable Equation :
4.4 Solve : x+1 = 0
Subtract 1 from both sides of the equation :
x = -1
Supplement : Solving Quadratic Equation Directly
Solving x2+10x+9 = 0 directly
Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula