**Solving Cubic Equations Word Problems :**

Here we are going to see some practice problems on solving cubic equations.

**Question 1 :**

If α, β and γ are the roots of the polynomial equation ax^{3} + bx^{2} + cx + d = 0 , find the Value of ∑ α/βγ in terms of the coefficients

**Solution :**

∑ α/βγ = (α/βγ) + (β/γα) + (γ/αβ)

= (α^{2}+ β^{2 }+ γ^{2})/αβγ

= (α + β^{ }+ γ)^{2 }- 2 (αβ+ βγ + αγ) ----(1)

α + β^{ }+ γ = -b/a

αβ+ βγ + αγ = c/a

By applying the above values in (1), we get

= (-b/a)^{2 }- 2 (c/a)

= (b^{2} - 2ca)/a^{2}

Hence the answer is (b^{2} - 2ca)/a^{2}.

**Question 2 :**

If α, β, γ and δ are the roots of the polynomial equation 2x^{4} + 5x^{3} − 7x^{2} + 8 = 0 , find a quadratic equation with integer coefficients whose roots are α + β + γ + δ and αβγδ.

**Solution :**

2x^{4} + 5x^{3} − 7x^{2} + 8 = 0

By comparing the given equation with the general form of polynomial of degree 4, we get

a = 2, b = 5, c = -7, d = 0 and e = 8.

α + β + γ + δ = -b/a = -5/2

αβγδ = e/a = 8/2 = 4

Let us look into the next problem on "Solving Cubic Equations Word Problems".

**Question 3 :**

If p and q are the roots of the equation lx^{2} + nx + n = 0, show that √(p/q) + √(q/p) + √(n/l) = 0

**Solution :**

Since p and q are the roots of the equation, let us find sum of roots and product of roots of the given quadratic equation.

Sum of roots = p + q = -n/l

Product of roots pq = n/l

L.H.S

√(p/q) + √(q/p) + √(n/l)

= (p + q)/√(pq) + √(n/l)

= (-n/l) / √(n/l) + √(n/l)

= - √(n/l) + √(n/l)

= 0

R.H.S

Hence proved.

Let us look into the next problem on "Solving Cubic Equations Word Problems".

**Question 4 :**

If the equations x^{2} + px + q = 0 and x^{2} + p'x + q' = 0 have a common root, show that it must be equal to (pq'-p'q)/q q' or (q - q') / (p' - p)

**Solution :**

Let "α" be the common root

By applying "α" instead of "x", we get

α^{2} + pα + q = 0 -----(1)

α^{2} + p'α + q' = 0 ------(2)

(1) - (2)

(pα + q) - (p'α + q') = 0

pα + q - p'α - q' = 0

(p - p')α + (q - q') = 0

α = - (q - q')/(p - p')

α = (q - q')/(p' - p)

To get the value of other root,

(1) x p' ==> p'α^{2} + pp'α + p'q = 0

(2) x p ==> p α^{2} + p'p α + q'p = 0

By subtracting these two equations, we get

(p'α^{2 }+ p'q) - (p α^{2} + q'p) = 0

p'α^{2 }+ p'q - p α^{2} - q'p = 0

α^{2}(p'- p) + (p'q - q'p) = 0

α^{2 }= (p'q - q'p) / (p'- p)

α^{ }= (p'q - q'p) / (p'- p) α

α^{ }= (p'q - q'p) / (q - q')

Hence proved.

Let us look into the next problem on "Solving Cubic Equations Word Problems".

**Question 5 :**

Formalate into a mathematical problem to find a number such that when its cube root is added to it, the result is 6.

**Solution :**

Let "x" be a required number.

Its cube root = x^{1/3}

x^{1/3 }+ x = 6

x^{1/3 } = (6 - x)

Taking cubes on both sides, we get

x^{ } = (6 - x)^{3}

By applying the algebraic identity for (a - b)^{3}, we get

(a - b)^{3} = a^{3} - 3a^{2} b + 3ab^{2} - b^{3}

x^{ } = 6^{3} - 3(36)x + 3(6)x^{2} - x^{3}

x^{ } = 216 - 108x + 18x^{2} - x^{3}

x^{3 }- 18x^{2} - 108x - x - 216 = 0

^{ }x^{3 }- 18x^{2} - 109x - 216 = 0

**Question 6 :**

A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was cut away

**Solution :**

Let "x" be the broken part, its cube root = x^{1/3}

x^{1/3 }+ x = 12

x^{1/3 } = (12 - x)

Taking cubes on both sides, we get

x^{ } = (12 - x)^{3}

By applying the algebraic identity for (a - b)^{3}, we get

(a - b)^{3} = a^{3} - 3a^{2} b + 3ab^{2} - b^{3}

x^{ } = 12^{3} - 3(144)x + 3(12)x^{2} - x^{3}

x^{ } = 1728 - 432x + 36x^{2} - x^{3}

x^{3 }- 36x^{2} + 432x + x - 1728 = 0

x^{3 }- 36x^{2} + 433x - 1728 = 0

x^{3 }+ x = 12

x^{3 }+ x - 12 = 0

After having gone through the stuff given above, we hope that the students would have understood, "Solving Cubic Equations Word Problems".

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**