Please solve the following problems: 1. If 3x+5=20, find the value of x. 2. Solve: 2(x−3)+5=17. 3. Factorise…

Question

Please solve the following problems: 1. If 3x+5=20, find the value of x. 2. Solve: 2(x−3)+5=17. 3. Factorise: x2-5x+6. 4. Expand and simplify: (x+2)(x−3).

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here are the solutions for all the problems from the image.

: If $3x+5=20$ , find the value of $x$ .

Subtract 5 from both sides of the equation.
$3x+5-5 = 20-5$ $3x = 15$
Divide both sides by 3.
$\frac{3x}{3} = \frac{15}{3}$ $x = 5$
The value of $x$ is:
$\boxed{5}$
: Solve: $2(x-3)+5=17$ .
Distribute the 2 into the parenthesis.
$2x - 6 + 5 = 17$
Combine the constant terms on the left side.
$2x - 1 = 17$
Add 1 to both sides of the equation.
$2x - 1 + 1 = 17 + 1$ $2x = 18$
Divide both sides by 2.
$\frac{2x}{2} = \frac{18}{2}$ $x = 9$
The value of $x$ is:
$\boxed{9}$
: Factorise: $x^2-5x+6$ .
Find two numbers that multiply to 6 and add up to -5.
The numbers are -2 and -3, since $(-2) \times (-3) = 6$ and $(-2) + (-3) = -5$ .
Write the quadratic expression in factored form using these numbers.
$(x-2)(x-3)$
The factored form is:
$\boxed{(x-2)(x-3)}$
: Expand and simplify: $(x+2)(x-3)$ .
Use the FOIL method (First, Outer, Inner, Last) to expand the product.
$(x+2)(x-3) = x \cdot x + x \cdot (-3) + 2 \cdot x + 2 \cdot (-3)$ $= x^2 - 3x + 2x - 6$
Combine the like terms.
$= x^2 - x - 6$
The expanded and simplified expression is:
$\boxed{x^2-x-6}$
: Solve the equation: $x^2-4x-12=0$ .
Factor the quadratic expression.
Find two numbers that multiply to -12 and add up to -4.
The numbers are -6 and 2, since $(-6) \times 2 = -12$ and $(-6) + 2 = -4$ .
$(x-6)(x+2) = 0$
Set each factor equal to zero and solve for $x$ .
$x-6 = 0 \quad \text{or} \quad x+2 = 0$ $x = 6 \quad \text{or} \quad x = -2$
The solutions for $x$ are:
$\boxed{x=6 \text{ or } x=-2}$
: If $\sin \theta = \frac{3}{5}$ , find the value of $\cos \theta$ .
Use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ .
Substitute the given value of $\sin \theta$ :
$\left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1$ $\frac{9}{25} + \cos^2 \theta = 1$
Subtract $\frac{9}{25}$ from both sides.
$\cos^2 \theta = 1 - \frac{9}{25}$ $\cos^2 \theta = \frac{25}{25} - \frac{9}{25}$ $\cos^2 \theta = \frac{16}{25}$
Take the square root of both sides.
$\cos \theta = \pm\sqrt{\frac{16}{25}}$ $\cos \theta = \pm\frac{4}{5}$
Assuming $\theta$ is an acute angle (in the first quadrant), $\cos \theta$ is positive.
The value of $\cos \theta$ is:
$\boxed{\frac{4}{5}}$
: In $\triangle ABC$ , $\angle A=50^\circ$ , $\angle B=60^\circ$ . Find $\angle C$ .
The sum of the angles in a triangle is $180^\circ$ .
$\angle A + \angle B + \angle C = 180^\circ$
Substitute the given values for $\angle A$ and $\angle B$ .
$50^\circ + 60^\circ + \angle C = 180^\circ$ $110^\circ + \angle C = 180^\circ$
Subtract $110^\circ$ from both sides to find $\angle C$ .
$\angle C = 180^\circ - 110^\circ$ $\angle C = 70^\circ$
The measure of $\angle C$ is:
$\boxed{70^\circ}$
: Find the area of a triangle with base 10 cm and height 6 cm.
Use the formula for the area of a triangle
Area $= \frac{1}{2} \times \text{base} \times \text{height}$ .
$\text{Area} = \frac{1}{2} \times 10 \text{ cm} \times 6 \text{ cm}$
Calculate the area.
$\text{Area} = \frac{1}{2} \times 60 \text{ cm}^2$ $\text{Area} = 30 \text{ cm}^2$
The area of the triangle is:
$\boxed{30 \text{ cm}^2}$
: The perimeter of a rectangle is 40 cm and its length is 14 cm. Find its breadth.
Use the formula for the perimeter of a rectangle
$P = 2(l+b)$ , where $P$ is the perimeter, $l$ is the length, and $b$ is the breadth.
Substitute the given values: $P=40$ cm and $l=14$ cm.
$40 = 2(14+b)$
Divide both sides by 2.
$\frac{40}{2} = 14+b$ $20 = 14+b$
Subtract 14 from both sides to find $b$ .
$20 - 14 = b$ $b = 6 \text{ cm}$
The breadth of the rectangle is:
$\boxed{6 \text{ cm}}$
: A bag contains 3 red balls, 4 blue balls and 5 green balls. One ball is drawn at random. What is the probability of getting a blue ball?
Calculate the total number of balls in the bag.
$\text{Total balls} = 3 (\text{red}) + 4 (\text{blue}) + 5 (\text{green}) = 12 \text{ balls}$
Identify the number of blue balls.
$\text{Number of blue balls} = 4$
Calculate the probability of drawing a blue ball using the formula
$P(\text{blue ball}) = \frac{\text{Number of blue balls}}{\text{Total number of balls}}$ $P(\text{blue ball}) = \frac{4}{12}$
Simplify the fraction.
$P(\text{blue ball}) = \frac{1}{3}$
The probability of getting a blue ball is:
$\boxed{\frac{1}{3}}$
: Simplify: $\left(\frac{2x}{3}\right)^2$ .
Apply the square to both the numerator and the denominator.
$\left(\frac{2x}{3}\right)^2 = \frac{(2x)^2}{3^2}$
Calculate the squares.
$= \frac{2^2 x^2}{3^2}$ $= \frac{4x^2}{9}$
The simplified expression is:
$\boxed{\frac{4x^2}{9}}$
: Find the value of $y$ if $\frac{y}{2}+3=11$ .
Subtract 3 from both sides of the equation.
$\frac{y}{2}+3-3 = 11-3$ $\frac{y}{2} = 8$
Multiply both sides by 2.
$\frac{y}{2} \times 2 = 8 \times 2$ $y = 16$
The value of $y$ is:
$\boxed{16}$
: Find the coordinates of the midpoint of the line segment joining $(2,3)$ and $(6,7)$ .
Use the midpoint formula
$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ .
Let $(x_1, y_1) = (2,3)$ and $(x_2, y_2) = (6,7)$ .
Substitute the coordinates into the formula.
$M = \left(\frac{2+6}{2}, \frac{3+7}{2}\right)$
Calculate the sums and divide by 2.
$M = \left(\frac{8}{2}, \frac{10}{2}\right)$ $M = (4,5)$
The coordinates of the midpoint are:
$\boxed{(4,5)}$
: In the figure, lines $l$ and $m$ are parallel. Find the value of $x$ .
Identify the relationship between the angles.
The angle $120^\circ$ and the angle adjacent to it on line $l$ form a linear pair, so their sum is $180^\circ$ . Let this adjacent angle be $\alpha$ .
$\alpha + 120^\circ = 180^\circ$ $\alpha = 180^\circ - 120^\circ = 60^\circ$
Recognize that $\alpha$ and $x^\circ$ are corresponding angles because lines $l$ and $m$ are parallel.
Corresponding angles are equal.
$x^\circ = \alpha$ $x^\circ = 60^\circ$
The value of $x$ is:
$\boxed{60}$
: Prove that: $(a+b)^2 = a^2+2ab+b^2$ .
Start with the left-hand side of the equation.
$(a+b)^2$
Expand the square as a product of two binomials.
$(a+b)^2 = (a+b)(a+b)$
Use the distributive property (FOIL method) to multiply the binomials.
$(a+b)(a+b) = a \cdot a + a \cdot b + b \cdot a + b \cdot b$ $= a^2 + ab + ba + b^2$
Combine the like terms ( $ab$ and $ba$ are the same).
$= a^2 + 2ab + b^2$
This matches the right-hand side of the equation, thus the identity is proven.
$\boxed{(a+b)^2 = a^2+2ab+b^2}$
All done

Please solve the following problems: 1. If 3x+5=20, find the value of x. 2. Solve: 2(x−3)+5=17. 3. Factorise: x2-5x+6. 4. Expand and simplify: (x+2)(x-3).

Step-by-step solution

Subtract 5 from both sides of the equation.

Divide both sides by 3.

Distribute the 2 into the parenthesis.

Combine the constant terms on the left side.

Add 1 to both sides of the equation.

Divide both sides by 2.

Find two numbers that multiply to 6 and add up to -5.

Write the quadratic expression in factored form using these numbers.

Use the FOIL method (First, Outer, Inner, Last) to expand the product.

Combine the like terms.

Factor the quadratic expression.

Set each factor equal to zero and solve for xxx.

Use the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1sin2θ+cos2θ=1.

Subtract 925\frac{9}{25}259​ from both sides.

Take the square root of both sides.

The sum of the angles in a triangle is 180∘180^\circ180∘.

Substitute the given values for ∠A\angle A∠A and ∠B\angle B∠B.

Subtract 110∘110^\circ110∘ from both sides to find ∠C\angle C∠C.

Use the formula for the area of a triangle

Calculate the area.

Use the formula for the perimeter of a rectangle

Divide both sides by 2.

Subtract 14 from both sides to find bbb.

Calculate the total number of balls in the bag.

Identify the number of blue balls.

Calculate the probability of drawing a blue ball using the formula

Simplify the fraction.

Apply the square to both the numerator and the denominator.

Calculate the squares.

Subtract 3 from both sides of the equation.

Multiply both sides by 2.

Use the midpoint formula

Substitute the coordinates into the formula.

Calculate the sums and divide by 2.

Identify the relationship between the angles.

Recognize that α\alphaα and x∘x^\circx∘ are corresponding angles because lines lll and mmm are parallel.

Start with the left-hand side of the equation.

Expand the square as a product of two binomials.

Use the distributive property (FOIL method) to multiply the binomials.

Combine the like terms (ababab and bababa are the same).

Still stuck on this one?

More Mathematics Questions

Set each factor equal to zero and solve for $x$ .

Use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ .

Subtract $\frac{9}{25}$ from both sides.

The sum of the angles in a triangle is $180^\circ$ .

Substitute the given values for $\angle A$ and $\angle B$ .

Subtract $110^\circ$ from both sides to find $\angle C$ .

Subtract 14 from both sides to find $b$ .

Recognize that $\alpha$ and $x^\circ$ are corresponding angles because lines $l$ and $m$ are parallel.

Combine the like terms ( $ab$ and $ba$ are the same).