Solve the following algebraic equations 4.1 x+2=0 4.2 (x−5)(x+2)=0 4.3 x2-25=0

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the questions:

Question 6.1.1
Determine the sum of $3x^2 + 7x + 7$ and $2x^2 + 3x + 4$ by adding like terms.

Write the sum of the two expressions.
$(3x^2 + 7x + 7) + (2x^2 + 3x + 4)$
Group like terms.
$(3x^2 + 2x^2) + (7x + 3x) + (7 + 4)$
Add the coefficients of like terms.
$5x^2 + 10x + 11$
The sum is $\boxed{\mathbf{5x^2 + 10x + 11}}$ .

Question 6.1.2
Subtract $2x^2 + 3x + 4$ from $3x^2 + 7x + 7$ .
Write the subtraction expression.
$(3x^2 + 7x + 7) - (2x^2 + 3x + 4)$
Distribute the negative sign and group like terms.
$3x^2 + 7x + 7 - 2x^2 - 3x - 4$ $(3x^2 - 2x^2) + (7x - 3x) + (7 - 4)$
Combine like terms.
$x^2 + 4x + 3$
The difference is $\boxed{\mathbf{x^2 + 4x + 3}}$ .

Question 6.2.1
Expand and simplify $(x-2)(x+2)$ .
Apply the difference of squares formula, $(a-b)(a+b) = a^2 - b^2$ .
$x^2 - 2^2$
Simplify the expression.
$x^2 - 4$
The expanded and simplified expression is $\boxed{\mathbf{x^2 - 4}}$ .

Question 6.2.2
Expand and simplify $(x+3)^2$ .
Apply the perfect square formula, $(a+b)^2 = a^2 + 2ab + b^2$ .
$x^2 + 2(x)(3) + 3^2$
Simplify the terms.
$x^2 + 6x + 9$
The expanded and simplified expression is $\boxed{\mathbf{x^2 + 6x + 9}}$ .

Question 6.3.1
Factorise $4x^2 - 12x$ .
Find the greatest common factor (GCF) of the terms.
The GCF of $4x^2$ and $12x$ is $4x$ .
Factor out the GCF.
$4x(x - 3)$
The factorised expression is $\boxed{\mathbf{4x(x - 3)}}$ .

Question 6.3.2
Factorise $x^2 + 4x + 3$ .
Find two numbers that multiply to 3 and add to 4.
These numbers are 1 and 3.
Write the quadratic as a product of two binomials.
$(x+1)(x+3)$
The factorised expression is $\boxed{\mathbf{(x+1)(x+3)}}$ .

Question 6.3.3
Factorise $x^2 - 4$ .
Recognize this as a difference of squares, $a^2 - b^2 = (a-b)(a+b)$ .
$x^2 - 2^2$
Apply the difference of squares formula.
$(x-2)(x+2)$
The factorised expression is $\boxed{\mathbf{(x-2)(x+2)}}$ .

Question 7.1.1
What are missing values of $x$ and $y$ on a flow diagram? The rule is $y = x-2$ .
For the missing output $y$ when input $x=2$
$y = 2 - 2$ $y = 0$
For the missing input $x$ when output $y=-1$
$-1 = x - 2$ $-1 + 2 = x$ $x = 1$
The missing values are $\boxed{\mathbf{x=1 \text{ and } y=0}}$ .

Question 7.1.2
Complete the table below. The rule is $y = x-2$ .
Apply the rule $y=x-2$ for each given $x$ value to find the corresponding $y$ value.
For $x = -1$ : $y = -1 - 2 = -3$
For $x = 0$ : $y = 0 - 2 = -2$
For $x = 2$ : $y = 2 - 2 = 0$
For $x = 4$ : $y = 4 - 2 = 2$
The completed table is
$\begin{array}{|c|c|c|c|c|} \hline x & -1 & 0 & 2 & 4 \\ \hline y = x-2 & -3 & -2 & 0 & 2 \\ \hline \end{array}$
The completed values for the table are $\boxed{\mathbf{y=2 \text{ for } x=4}}$ .

Question 7.1.3
Describe the rule that is used to find the output of the function plane.

The rule is $\boxed{\mathbf{y = x-2}}$ . This means to find the output, you subtract 2 from the input value.

Question 6.4.1
Solve $x+2=4$ .
Subtract 2 from both sides of the equation.
$x + 2 - 2 = 4 - 2$
Simplify the equation.
$x = 2$
The solution is $\boxed{\mathbf{x=2}}$ .

Question 6.4.2
Solve $(x-5)(x+2)=0$ .
Set each factor equal to zero.
$x-5 = 0 \quad \text{or} \quad x+2 = 0$
Solve each linear equation for $x$ .
$x = 5 \quad \text{or} \quad x = -2$
The solutions are $\boxed{\mathbf{x=5 \text{ or } x=-2}}$ .

Question 6.4.3
Solve $x^2-25=0$ .
Add 25 to both sides of the equation.
$x^2 = 25$
Take the square root of both sides, remembering both positive and negative roots.
$x = \pm\sqrt{25}$ $x = \pm 5$
The solutions are $\boxed{\mathbf{x=5 \text{ or } x=-5}}$ .

Solve the following algebraic equations 4.1 x+2=0 4.2 (x−5)(x+2)=0 4.3 x2−25=0

Step-by-step solution

Write the sum of the two expressions.

Group like terms.

Add the coefficients of like terms.

Write the subtraction expression.

Distribute the negative sign and group like terms.

Combine like terms.

Apply the difference of squares formula, $(a-b)(a+b) = a^2 - b^2$ .

Simplify the expression.

Apply the perfect square formula, $(a+b)^2 = a^2 + 2ab + b^2$ .

Simplify the terms.

Find the greatest common factor (GCF) of the terms.

Factor out the GCF.

Find two numbers that multiply to 3 and add to 4.

Write the quadratic as a product of two binomials.

Recognize this as a difference of squares, $a^2 - b^2 = (a-b)(a+b)$ .

Apply the difference of squares formula.

For the missing output $y$ when input $x=2$

For the missing input $x$ when output $y=-1$

Apply the rule $y=x-2$ for each given $x$ value to find the corresponding $y$ value.

The completed table is

Subtract 2 from both sides of the equation.

Simplify the equation.

Set each factor equal to zero.

Solve each linear equation for $x$ .

Add 25 to both sides of the equation.

Take the square root of both sides, remembering both positive and negative roots.

Still stuck on this one?

More Mathematics Questions