Solve the equations by completing the square.
You will need to work them out on paper.
Give exact answers for these questions:
$$
\begin{array}{l}
x^{2}+16 x+5=0 \
x^{2}-14 x+10=0 \
x= \
\text { or } x= \
-\sqrt{ } \
+\sqrt{ } \
\text { or } x= \
-\sqrt{ }
\end{array}
$$

Give your answers to $1 \mathrm{~d} . \mathrm{p}$. or better for these questions:
$$
\begin{array}{l}
x^{2}-5 x=10 \
x=\square
\end{array}
$$
or $x=$ $\square$
$$
6 x^{2}-54 x+30=0
$$
$x=$ $\square$ or $x=$ $\square$
$$
5 x^{2}-60 x+30=0 \quad 2 x^{2}+16 x=5
$$
$x=$ $\square$ or $x=$ $\square$
$x=$ $\square$ or $x=$ $\square$
[12]

Question

Solve the equations by completing the square.
You will need to work them out on paper.
Give exact answers for these questions:
$$
\begin{array}{l}
x^{2}+16 x+5=0 \
x^{2}-14 x+10=0 \
x= \
\text { or } x= \
-\sqrt{ } \
+\sqrt{ } \
\text { or } x= \
-\sqrt{ }
\end{array}
$$

Give your answers to $1 \mathrm{~d} . \mathrm{p}$. or better for these questions:
$$
\begin{array}{l}
x^{2}-5 x=10 \
x=\square
\end{array}
$$
or $x=$ $\square$
$$
6 x^{2}-54 x+30=0
$$
$x=$ $\square$ or $x=$ $\square$
$$
5 x^{2}-60 x+30=0 \quad 2 x^{2}+16 x=5
$$
$x=$ $\square$ or $x=$ $\square$
$x=$ $\square$ or $x=$ $\square$
[12]

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To solve the equation $x^2 + 16x + 5 = 0$, we can complete the square. First, we rewrite it as $x^2 + 16x = -5$. ‖ ‖ step 2 ⋮ Next, we take half of the coefficient of $x$, which is $8$, and square it to get $64$. We add $64$ to both sides: $x^2 + 16x + 64 = 59$. ‖ ‖ step 3 ⋮ Now, we can factor the left side: $(x + 8)^2 = 59$. Taking the square root of both sides gives us $x + 8 = \pm \sqrt{59}$. ‖ ‖ step 4 ⋮ Finally, we solve for $x$: $x = -8 \pm \sqrt{59}$. ‖ ‖ step 5 ⋮ For the second equation $x^2 - 14x + 10 = 0$, we rewrite it as $x^2 - 14x = -10$. ‖ ‖ step 6 ⋮ Taking half of the coefficient of $x$, which is $-7$, and squaring it gives us $49$. We add $49$ to both sides: $x^2 - 14x + 49 = 39$. ‖ ‖ step 7 ⋮ Factoring gives us $(x - 7)^2 = 39$. Taking the square root of both sides gives us $x - 7 = \pm \sqrt{39}$. ‖ ‖ step 8 ⋮ Finally, we solve for $x$: $x = 7 \pm \sqrt{39}$. ‖ ‖ step 9 ⋮ For the equation $x^2 - 5x = 10$, we rewrite it as $x^2 - 5x - 10 = 0$. ‖ ‖ step 10 ⋮ Completing the square gives us $x^2 - 5x + \frac{25}{4} = \frac{65}{4}$. ‖ ‖ step 11 ⋮ Factoring gives us $(x - \frac{5}{2})^2 = \frac{65}{4}$. Taking the square root gives us $x - \frac{5}{2} = \pm \frac{\sqrt{65}}{2}$. ‖ ‖ step 12 ⋮ Finally, we solve for $x$: $x = \frac{5 \pm \sqrt{65}}{2}$. ‖ ‖ step 13 ⋮ For the equation $6x^2 - 54x + 30 = 0$, we divide by 6 to simplify: $x^2 - 9x + 5 = 0$. ‖ ‖ step 14 ⋮ Completing the square gives us $x^2 - 9x + \frac{81}{4} = \frac{89}{4}$. ‖ ‖ step 15 ⋮ Factoring gives us $(x - \frac{9}{2})^2 = \frac{89}{4}$. Taking the square root gives us $x - \frac{9}{2} = \pm \frac{\sqrt{89}}{2}$. ‖ ‖ step 16 ⋮ Finally, we solve for $x$: $x = \frac{9 \pm \sqrt{89}}{2}$. ‖ ‖ step 17 ⋮ For the equation $5x^2 - 60x + 30 = 0$, we divide by 5 to simplify: $x^2 - 12x + 6 = 0$. ‖ ‖ step 18 ⋮ Completing the square gives us $x^2 - 12x + 36 = 30$. ‖ ‖ step 19 ⋮ Factoring gives us $(x - 6)^2 = 30$. Taking the square root gives us $x - 6 = \pm \sqrt{30}$. ‖ ‖ step 20 ⋮ Finally, we solve for $x$: $x = 6 \pm \sqrt{30}$. ‖ ‖ step 21 ⋮ For the equation $2x^2 + 16x = 5$, we rewrite it as $2x^2 + 16x - 5 = 0$. ‖ ‖ step 22 ⋮ Completing the square gives us $2(x^2 + 8x + 16) = 36$. ‖ ‖ step 23 ⋮ Factoring gives us $2(x + 4)^2 = 36$. Taking the square root gives us $x + 4 = \pm 3$. ‖ ‖ step 24 ⋮ Finally, we solve for $x$: $x = -4 \pm 3$. ‖ ∻Answer∻ ⚹ For the equations: 1. $x = -8 \pm \sqrt{59}$ 2. $x = 7 \pm \sqrt{39}$ 3. $x = \frac{5 \pm \sqrt{65}}{2}$ 4. $x = \frac{9 \pm \sqrt{89}}{2}$ 5. $x = 6 \pm \sqrt{30}$ 6. $x = -4 \pm 3$ ⚹ ∻Key Concept∻ ⚹Completing the square is a method used to solve quadratic equations by rewriting them in a perfect square form.⚹ ∻Explanation∻ ⚹The solutions are derived by transforming the quadratic equations into a form that allows us to easily extract the roots. Each step involves manipulating the equation to isolate the variable.⚹◊What is the general form of a quadratic equation, and how can you identify the coefficients to complete the square?⍭ Generate me a similar question◊

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