Question

How do you simplify \[5{z^2} + 3z + 8{z^2}\]?

Answer 1

Hint: Solve the given equation i.e. bring all coefficients together for the same variables. Equate the equation to 0 in order to simplify the equation. Use a method of determinant to solve for the value of x from the given quadratic equation. Compare the quadratic equation with general quadratic equation and substitute values in the formula of finding roots of the equation.
* For a general quadratic equation \[a{x^2} + bx + c = 0\], roots are given by formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]

Complete step-by-step solution:
We are given the quadratic equation \[5{z^2} + 3z + 8{z^2}\]
Since we have two values for same variable we can add like terms i.e. \[(5 + 8){z^2} + 3z\]
The equation becomes \[13{z^2} + 3z\]
To simplify the equation, we will equate the equation to 0
We will solve \[13{z^2} + 3z = 0\]...........… (1)
We know that general quadratic equation is \[a{z^2} + bz + c = 0\]
On comparing with general quadratic equation \[a{z^2} + bz + c = 0\], we get \[a = 13,b = 3,c = 0\]
Substitute the values of a, b and c in the formula of finding roots of the equation.
\[ \Rightarrow z = \dfrac{{ - (3) \pm \sqrt {{{(3)}^2} - 4 \times 13 \times 0} }}{{2 \times 13}}\]
Square the values inside the square root in numerator of the fraction
\[ \Rightarrow z = \dfrac{{ - 3 \pm \sqrt {{3^2}} }}{{2 \times 13}}\]
Cancel square root by square power in the numerator
\[ \Rightarrow z = \dfrac{{ - 3 \pm 3}}{{2 \times 13}}\]
So, \[z = \dfrac{{ - 3 + 3}}{{2 \times 13}}\] and \[z = \dfrac{{ - 3 - 3}}{{2 \times 13}}\]
I.e. \[z = \dfrac{0}{{2 \times 13}}\] and \[z = \dfrac{{ - 6}}{{2 \times 13}}\]
Cancel possible factors from numerator and denominator
\[z = 0\]and \[z = \dfrac{{ - 3}}{{13}}\]

\[\therefore \]Solution of the equation \[5{z^2} + 3z + 8{z^2}\] is \[z = 0\] and \[z = \dfrac{{ - 3}}{{13}}\].

Note: Alternate method:
We have to simplify the equation \[5{z^2} + 3z + 8{z^2}\]
We can write \[5{z^2} + 3z + 8{z^2} = 13{z^2} + 3z\]
To simplify an equation we equate it to 0
\[ \Rightarrow 13{z^2} + 3z = 0\]
Take z common from both terms
\[ \Rightarrow z(13z + 3) = 0\]
We know products with two values can be zero when one of them is 0 or both are zero. We equate both values to 0.
\[ \Rightarrow z = 0\] and \[13z + 3 = 0\]
Shift constant to RHS
\[ \Rightarrow z = 0\] and \[13z = - 3\]
Cross multiply the value from LHS to denominator of RHS
\[ \Rightarrow z = 0\] and \[z = \dfrac{{ - 3}}{{13}}\]
\[\therefore \]Solution of the equation \[5{z^2} + 3z + 8{z^2}\] is \[z = 0\] and \[z = \dfrac{{ - 3}}{{13}}\].