Q: How many tablespoons are equivalent to 3 ounces?

The correct answer is:6 tablespoons Explanation A US fluid ounce is a measure of volume: $1\ \text{fl oz} = 2\ \text{tbsp}$. So multiply that conversion by 3. Steps: Use the conversion: $$1\ \text{fl oz} = 2\ \text{tbsp}$$ Multiply by 3: $$3\ \text{fl oz} = 3 \times 2\ \text{tbsp} = 6\ \text{tbsp}$$ Therefore, 3 ounces equals 6 tablespoons.

Q: How many pounds is 62 kg?

The correct answer is: $136.6866025546241\text{ lb}$ Explanation We use the conversion factor $1\text{ kg} = 2.2046226218487757\text{ lb}$. Steps: Multiply the mass by the conversion factor: $$62\text{ kg}\times 2.2046226218487757\frac{\text{lb}}{\text{kg}} = 136.6866025546241\text{ lb}$$ Round as desired: to two decimal places $$\approx 136.69\text{ lb}$$ (to one decimal place $$\approx 136.7\text{ lb}$$). Therefore, 62 kg ≈ 136.69 lb (exactly about 136.6866 lb).

Q: Convert 165 pounds into kilograms.

The correct answer is: $74.84274105\text{ kg}$ Explanation We convert pounds to kilograms using the exact conversion factor $1\text{ lb} = 0.45359237\text{ kg}$. Multiplying 165 pounds by that factor gives the mass in kilograms. Steps: Use the conversion factor: $1\text{ lb} = 0.45359237\text{ kg}$. Multiply: $$165\times 0.45359237 = 74.84274105$$ Attach the unit: $74.84274105\text{ kg}$. Therefore, 165 pounds equals $74.84274105\text{ kg}$ (commonly rounded to $74.84\text{ kg}$).

Q: What is 300 milliliters in fluid ounces?

The correct answer is: $10.14\text{ US fl oz (approx.)}$ Explanation We convert using the US fluid ounce definition: $1\text{ US fl oz} = 29.5735295625\text{ mL}$, so multiply mL by the factor $1/29.5735295625 \approx 0.0338140226$. Steps: Multiply: $$300\text{ mL}\times 0.0338140226\frac{\text{US fl oz}}{\text{mL}}=10.14420678\text{ US fl oz}$$ Round to a sensible precision: $$\approx10.14\text{ US fl oz}$$ If you meant imperial (UK) fluid ounces instead: $$300\text{ mL}\div28.4130625\frac{\text{mL}}{\text{imp fl oz}}\approx10.56\text{ imp fl oz}$$

Q: What is 103 kg in lbs?

The correct answer is: $227.08\text{ lb}$ Explanation To convert kilograms to pounds multiply by the conversion factor $2.2046226218\text{ lb/kg}$. For 103 kg, multiply 103 by that factor and round to a convenient number of decimal places. Steps: Conversion factor: $1\text{ kg} = 2.2046226218\text{ lb}$ Multiply: $103\text{ kg}\times 2.2046226218\text{ lb/kg} = 227.0761300454\text{ lb}$ Round: $227.0761300454\text{ lb}\approx 227.08\text{ lb}$ Therefore, $103\text{ kg}\approx 227.08\text{ lb}$.

Question

Quadratic Equation | Definition, Formula & Examples

Accepted Answer

A quadratic equation is a second-degree polynomial equation in one variable, typically written as $ax^2 + bx + c = 0$, where $a \\neq 0$.

Explanation

A quadratic equation involves the squared term $x^2$ and can have up to two real roots. Key features include the coefficients $a,b,c$, the discriminant that determines root types, the vertex (maximum or minimum), and a parabolic graph.

Standard form

The standard form is:
$$ax^2 + bx + c = 0$$

Quadratic formula (roots)

The solutions (roots) are given by the quadratic formula:
$$x = \\frac{-b \\pm \\sqrt{b^2 – 4ac}}{2a}$$

Derivation (completing the square)

Starting from $ax^2 + bx + c = 0$ (with $a\\neq0$), divide by $a$:
$$x^2 + \\frac{b}{a}x + \\frac{c}{a} = 0$$
Move constant term:
$$x^2 + \\frac{b}{a}x = -\\frac{c}{a}$$
Add $(\\frac{b}{2a})^2$ to both sides:
$$x^2 + \\frac{b}{a}x + \\left(\\frac{b}{2a}\\right)^2 = -\\frac{c}{a} + \\left(\\frac{b}{2a}\\right)^2$$
Left side is a perfect square:
$$\\left(x + \\frac{b}{2a}\\right)^2 = \\frac{b^2 – 4ac}{4a^2}$$
Take square root and solve for $x$ to get the quadratic formula:
$$x = \\frac{-b \\pm \\sqrt{b^2 – 4ac}}{2a}$$

Discriminant and nature of roots

Define the discriminant $\Delta = b^2 – 4ac$.

If $\Delta > 0$: two distinct real roots.
If $\Delta = 0$: one real repeated root (double root).
If $\Delta < 0$: two complex conjugate roots.

Methods to solve

Factoring (when factors are integers or simple).
Completing the square (derivation method; useful for vertex form).
Quadratic formula (works always).
Graphing (visual; finds intercepts of parabola $y=ax^2+bx+c$).

Vertex form and graph

Vertex form: $$y = a(x – h)^2 + k$$ where vertex is $(h,k)$ and axis of symmetry is $x=h$. For $ax^2 + bx + c$, the vertex is at:
$$h = -\\frac{b}{2a},\\quad k = c – \\frac{b^2}{4a}$$
The graph is a parabola opening up if $a>0$ (minimum) and down if $a<0$ (maximum).

Worked examples

1) Factorable example:
Solve $x^2 – 5x + 6 = 0$.
Steps:

Factor: $(x-2)(x-3)=0$.
Roots: $x=2$ or $x=3$.

Therefore, the solutions are $x=2$ and $x=3$.

2) Using the quadratic formula:
Solve $x^2 + 4x + 1 = 0$.
Steps:

Here $a=1,b=4,c=1$. Compute discriminant: $\Delta = 4^2 – 4(1)(1)=12$.
Apply formula: $$x = \\frac{-4 \\pm \\sqrt{12}}{2} = \\frac{-4 \\pm 2\\sqrt{3}}{2} = -2 \\pm \\sqrt{3}.$$

Therefore, $x = -2 \\pm \\sqrt{3}$.

3) Completing the square:
Solve $2x^2 – 8x + 5 = 0$.
Steps:

Divide by 2: $x^2 – 4x + \\tfrac{5}{2}=0$.
Move constant: $x^2 – 4x = -\\tfrac{5}{2}$.
Add $(\\tfrac{4}{2})^2=4$: $x^2 -4x +4 = -\\tfrac{5}{2} +4 = \\tfrac{3}{2}$.
So $(x-2)^2 = \\tfrac{3}{2}$, $x-2 = \\pm \\sqrt{\\tfrac{3}{2}}$.
Solutions: $x = 2 \\pm \\sqrt{\\tfrac{3}{2}}$ (can rationalize if desired).

Applications & tips

Quadratics model projectile motion, area problems, optimization, and many physics/economics situations. If factoring fails quickly, use the quadratic formula. Check the discriminant first to anticipate the type of roots.

Practice problems

Solve $x^2 + 2x – 8 = 0$.
Find vertex and axis of symmetry for $y = -3x^2 + 6x -1$.
Solve $3x^2 + x + 5 = 0$ and classify the roots.

If you want, I can solve the practice problems step-by-step or make a short worksheet with solutions.